Imports
/-
Copyright (c) 2026 Google LLC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
-/
module
public import Cryptids.Riemann120
public import Cryptids.Semantics.Riemann.Schmidt
Arithmetic-to-Analytic Bridge & One-Sided Landau Converse for riemann_120
This standalone file establishes the number-theoretic and complex-analytic bridge between the
self-sieving base-n offset digit deficit (ShuniaCounterexample) and Mathlib's
RiemannHypothesis:
Arithmetic & Logarithmic Lemmas:
log_primorial_eq_theta: Real.log (primorial n) = Chebyshev.theta n
log_shuniaProd_eq: Real.log (shuniaProd n) = Real.log (n.factorial) + Chebyshev.theta n
factorial_centering_bounds: n * log n - n + 1 ≤ log (n!) ≤ n * log n - n + 1 + log n
shuniaProd_not_pow: shuniaProd n ≠ n ^ d for all n ≥ 3
pow_sub_one_le_of_offsetDigits and offsetDigits_pow_gt: sharp power sandwiches for
offsetDigits
shuniaDeficit_mul_log_le and self_sub_theta_le_shuniaDeficit_mul_log: two-sided bridge
between shuniaDeficit n * Real.log n and (n : ℝ) - Chebyshev.theta n
Quartering Clock & Unconditional Small-n Verification (2 ≤ n ≤ 21845):
quarterClock_spec:
4 ^ quarterClock (n - 1) ≤ 3 * n ∧ 3 * n < 4 ^ (quarterClock (n - 1) + 1)
not_shuniaCounterexample_of_le_21845: unconditional verification for all 2 ≤ n ≤ 21845
using factorial centering (n ≤ 1365) and Chebyshev's prime lower bound (1366 ≤ n ≤ 21845)
One-Sided Landau Converse (¬ RiemannHypothesis → ∃ n ≥ 2, ShuniaCounterexample n):
Analytic continuation and residue contradiction for psiNegDefect C α t = C * t ^ α + ψ(t) - t
not_riemannHypothesis_imp_exists_shuniaCounterexample
exists_shuniaCounterexample_iff_not_riemannHypothesis_of_large
@[expose] public sectionopen Complex Finset Filter MeasureTheory Asymptotics Set Topology RiemannSchmidtopen scoped Real Topology Nat ENNReal LSeries.notation ArithmeticFunctionnamespace Riemann120Analyticopen ConjectureMachines.Riemann120
(shuniaProd offsetDigits shuniaDeficit quarterClock ShuniaCounterexample)abbrev primorial : ℕ → ℕ := ConjectureMachines.Riemann120.primorial2. Arithmetic & Logarithmic Lemmas
theorem primorial_eq_root_primorial (n : ℕ) : primorial n = _root_.primorial n := n:ℕ⊢ primorial n = _root_.primorial n
n:ℕ⊢ (filter Nat.Prime (Finset.Icc 1 n)).prod id = ∏ p ∈ Finset.range (n + 1) with Nat.Prime p, p
n:ℕ⊢ filter Nat.Prime (Finset.Icc 1 n) = {p ∈ Finset.range (n + 1) | Nat.Prime p}
n:ℕp:ℕ⊢ p ∈ filter Nat.Prime (Finset.Icc 1 n) ↔ p ∈ {p ∈ Finset.range (n + 1) | Nat.Prime p}
n:ℕp:ℕ⊢ (1 ≤ p ∧ p ≤ n) ∧ Nat.Prime p ↔ p < n + 1 ∧ Nat.Prime p
exact ⟨fun ⟨⟨_, h2⟩, hp⟩ ↦ ⟨n:ℕp:ℕx✝:(1 ≤ p ∧ p ≤ n) ∧ Nat.Prime pleft✝:1 ≤ ph2:p ≤ nhp:Nat.Prime p⊢ p < n + 1 All goals completed! 🐙, hp⟩, fun ⟨h2, hp⟩ ↦ ⟨⟨hp.one_le, n:ℕp:ℕx✝:p < n + 1 ∧ Nat.Prime ph2:p < n + 1hp:Nat.Prime p⊢ p ≤ n All goals completed! 🐙⟩, hp⟩⟩theorem primorial_pos (n : ℕ) : 0 < primorial n := n:ℕ⊢ 0 < primorial n
n:ℕ⊢ 0 < _root_.primorial n
All goals completed! 🐙theorem shuniaProd_pos (n : ℕ) : 0 < shuniaProd n :=
Nat.mul_pos (Nat.factorial_pos n) (primorial_pos n)theorem four_le_shuniaProd {n : ℕ} (hn : 2 ≤ n) : 4 ≤ shuniaProd n := n:ℕhn:2 ≤ n⊢ 4 ≤ shuniaProd n
n:ℕhn:2 ≤ nhfac:2 ≤ n !⊢ 4 ≤ shuniaProd n
n:ℕhn:2 ≤ nhfac:2 ≤ n !hprim:2 ≤ primorial n⊢ 4 ≤ shuniaProd n
n:ℕhn:2 ≤ nhfac:2 ≤ n !hprim:2 ≤ (filter Nat.Prime (Finset.Icc 1 n)).prod id⊢ 4 ≤ n ! * (filter Nat.Prime (Finset.Icc 1 n)).prod id
All goals completed! 🐙
log(P(n)) = θ(n) for every n : ℕ.
theorem log_primorial_eq_theta (n : ℕ) : Real.log (primorial n) = Chebyshev.theta n := n:ℕ⊢ Real.log ↑(primorial n) = Chebyshev.theta ↑n
All goals completed! 🐙
log(A_n) = log(n!) + θ(n) for every n : ℕ.
theorem log_shuniaProd_eq (n : ℕ) :
Real.log (shuniaProd n) = Real.log (n.factorial) + Chebyshev.theta n := n:ℕ⊢ Real.log ↑(shuniaProd n) = Real.log ↑n ! + Chebyshev.theta ↑n
n:ℕ⊢ Real.log ↑(n ! * ConjectureMachines.Riemann120.primorial n) = Real.log ↑n ! + Chebyshev.theta ↑n
All goals completed! 🐙
Elementary step ((k + 1 : ℕ) : ℝ) * (Real.log k - Real.log (k + 1)) ≤ -1 for k ≥ 1.
lemma succ_mul_log_sub_log_succ_le_neg_one {k : ℕ} (hk : 1 ≤ k) :
((k + 1 : ℕ) : ℝ) * (Real.log k - Real.log (k + 1 : ℕ)) ≤ -1 := k:ℕhk:1 ≤ k⊢ ↑(k + 1) * (Real.log ↑k - Real.log ↑(k + 1)) ≤ -1
k:ℕhk:1 ≤ khk_pos:0 < ↑k⊢ ↑(k + 1) * (Real.log ↑k - Real.log ↑(k + 1)) ≤ -1
k:ℕhk:1 ≤ khk_pos:0 < ↑khk1_pos:0 < ↑(k + 1)⊢ ↑(k + 1) * (Real.log ↑k - Real.log ↑(k + 1)) ≤ -1
k:ℕhk:1 ≤ khk_pos:0 < ↑khk1_pos:0 < ↑(k + 1)hdiv_pos:0 < ↑k / ↑(k + 1)⊢ ↑(k + 1) * (Real.log ↑k - Real.log ↑(k + 1)) ≤ -1
k:ℕhk:1 ≤ khk_pos:0 < ↑khk1_pos:0 < ↑(k + 1)hdiv_pos:0 < ↑k / ↑(k + 1)hexp:-1 / ↑(k + 1) + 1 ≤ rexp (-1 / ↑(k + 1))⊢ ↑(k + 1) * (Real.log ↑k - Real.log ↑(k + 1)) ≤ -1
k:ℕhk:1 ≤ khk_pos:0 < ↑khk1_pos:0 < ↑(k + 1)hdiv_pos:0 < ↑k / ↑(k + 1)hexp:-1 / ↑(k + 1) + 1 ≤ rexp (-1 / ↑(k + 1))heq:-1 / ↑(k + 1) + 1 = ↑k / ↑(k + 1)⊢ ↑(k + 1) * (Real.log ↑k - Real.log ↑(k + 1)) ≤ -1
k:ℕhk:1 ≤ khk_pos:0 < ↑khk1_pos:0 < ↑(k + 1)hdiv_pos:0 < ↑k / ↑(k + 1)hexp:↑k / ↑(k + 1) ≤ rexp (-1 / ↑(k + 1))heq:-1 / ↑(k + 1) + 1 = ↑k / ↑(k + 1)⊢ ↑(k + 1) * (Real.log ↑k - Real.log ↑(k + 1)) ≤ -1
k:ℕhk:1 ≤ khk_pos:0 < ↑khk1_pos:0 < ↑(k + 1)hdiv_pos:0 < ↑k / ↑(k + 1)hexp:↑k / ↑(k + 1) ≤ rexp (-1 / ↑(k + 1))heq:-1 / ↑(k + 1) + 1 = ↑k / ↑(k + 1)hlog:Real.log (↑k / ↑(k + 1)) ≤ -1 / ↑(k + 1)⊢ ↑(k + 1) * (Real.log ↑k - Real.log ↑(k + 1)) ≤ -1
k:ℕhk:1 ≤ khk_pos:0 < ↑khk1_pos:0 < ↑(k + 1)hdiv_pos:0 < ↑k / ↑(k + 1)hexp:↑k / ↑(k + 1) ≤ rexp (-1 / ↑(k + 1))heq:-1 / ↑(k + 1) + 1 = ↑k / ↑(k + 1)hlog:Real.log (↑k / ↑(k + 1)) ≤ -1 / ↑(k + 1)hmul:↑(k + 1) * Real.log (↑k / ↑(k + 1)) ≤ -1⊢ ↑(k + 1) * (Real.log ↑k - Real.log ↑(k + 1)) ≤ -1
All goals completed! 🐙
Elementary step (k : ℝ) * (Real.log (k + 1) - Real.log k) ≤ 1 for k ≥ 1.
lemma mul_log_succ_sub_log_le_one {k : ℕ} (hk : 1 ≤ k) :
(k : ℝ) * (Real.log (k + 1 : ℕ) - Real.log k) ≤ 1 := k:ℕhk:1 ≤ k⊢ ↑k * (Real.log ↑(k + 1) - Real.log ↑k) ≤ 1
k:ℕhk:1 ≤ khk_pos:0 < ↑k⊢ ↑k * (Real.log ↑(k + 1) - Real.log ↑k) ≤ 1
k:ℕhk:1 ≤ khk_pos:0 < ↑khk1_pos:0 < ↑(k + 1)⊢ ↑k * (Real.log ↑(k + 1) - Real.log ↑k) ≤ 1
k:ℕhk:1 ≤ khk_pos:0 < ↑khk1_pos:0 < ↑(k + 1)hexp:1 / ↑k + 1 ≤ rexp (1 / ↑k)⊢ ↑k * (Real.log ↑(k + 1) - Real.log ↑k) ≤ 1
k:ℕhk:1 ≤ khk_pos:0 < ↑khk1_pos:0 < ↑(k + 1)hexp:1 / ↑k + 1 ≤ rexp (1 / ↑k)heq:1 / ↑k + 1 = ↑(k + 1) / ↑k⊢ ↑k * (Real.log ↑(k + 1) - Real.log ↑k) ≤ 1
k:ℕhk:1 ≤ khk_pos:0 < ↑khk1_pos:0 < ↑(k + 1)hexp:↑(k + 1) / ↑k ≤ rexp (1 / ↑k)heq:1 / ↑k + 1 = ↑(k + 1) / ↑k⊢ ↑k * (Real.log ↑(k + 1) - Real.log ↑k) ≤ 1
k:ℕhk:1 ≤ khk_pos:0 < ↑khk1_pos:0 < ↑(k + 1)hexp:↑(k + 1) / ↑k ≤ rexp (1 / ↑k)heq:1 / ↑k + 1 = ↑(k + 1) / ↑khlog:Real.log (↑(k + 1) / ↑k) ≤ 1 / ↑k⊢ ↑k * (Real.log ↑(k + 1) - Real.log ↑k) ≤ 1
k:ℕhk:1 ≤ khk_pos:0 < ↑khk1_pos:0 < ↑(k + 1)hexp:↑(k + 1) / ↑k ≤ rexp (1 / ↑k)heq:1 / ↑k + 1 = ↑(k + 1) / ↑khlog:Real.log (↑(k + 1) / ↑k) ≤ 1 / ↑khmul:↑k * Real.log (↑(k + 1) / ↑k) ≤ 1⊢ ↑k * (Real.log ↑(k + 1) - Real.log ↑k) ≤ 1
All goals completed! 🐙
Factorial centering bounds:
n * log n - n + 1 ≤ log(n!) ≤ n * log n - n + 1 + log n for all n ≥ 1.
theorem factorial_centering_bounds (n : ℕ) (hn : 1 ≤ n) :
(n : ℝ) * Real.log n - n + 1 ≤ Real.log (n.factorial) ∧
Real.log (n.factorial) ≤ (n : ℝ) * Real.log n - n + 1 + Real.log n := n:ℕhn:1 ≤ n⊢ ↑n * Real.log ↑n - ↑n + 1 ≤ Real.log ↑n ! ∧ Real.log ↑n ! ≤ ↑n * Real.log ↑n - ↑n + 1 + Real.log ↑n
induction n with
hn:1 ≤ 0⊢ ↑0 * Real.log ↑0 - ↑0 + 1 ≤ Real.log ↑0! ∧ Real.log ↑0! ≤ ↑0 * Real.log ↑0 - ↑0 + 1 + Real.log ↑0 All goals completed! 🐙
k:ℕih:1 ≤ k → ↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k ! ∧ Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khn:1 ≤ k + 1⊢ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 ≤ Real.log ↑(k + 1)! ∧
Real.log ↑(k + 1)! ≤ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 + Real.log ↑(k + 1)
ih:1 ≤ 0 → ↑0 * Real.log ↑0 - ↑0 + 1 ≤ Real.log ↑0! ∧ Real.log ↑0! ≤ ↑0 * Real.log ↑0 - ↑0 + 1 + Real.log ↑0hn:1 ≤ 0 + 1⊢ ↑(0 + 1) * Real.log ↑(0 + 1) - ↑(0 + 1) + 1 ≤ Real.log ↑(0 + 1)! ∧
Real.log ↑(0 + 1)! ≤ ↑(0 + 1) * Real.log ↑(0 + 1) - ↑(0 + 1) + 1 + Real.log ↑(0 + 1)k:ℕih:1 ≤ k → ↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k ! ∧ Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khn:1 ≤ k + 1hk_pos:0 < k⊢ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 ≤ Real.log ↑(k + 1)! ∧
Real.log ↑(k + 1)! ≤ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 + Real.log ↑(k + 1)
ih:1 ≤ 0 → ↑0 * Real.log ↑0 - ↑0 + 1 ≤ Real.log ↑0! ∧ Real.log ↑0! ≤ ↑0 * Real.log ↑0 - ↑0 + 1 + Real.log ↑0hn:1 ≤ 0 + 1⊢ ↑(0 + 1) * Real.log ↑(0 + 1) - ↑(0 + 1) + 1 ≤ Real.log ↑(0 + 1)! ∧
Real.log ↑(0 + 1)! ≤ ↑(0 + 1) * Real.log ↑(0 + 1) - ↑(0 + 1) + 1 + Real.log ↑(0 + 1) All goals completed! 🐙
k:ℕih:1 ≤ k → ↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k ! ∧ Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khn:1 ≤ k + 1hk_pos:0 < k⊢ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 ≤ Real.log ↑(k + 1)! ∧
Real.log ↑(k + 1)! ≤ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 + Real.log ↑(k + 1) k:ℕih:1 ≤ k → ↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k ! ∧ Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khn:1 ≤ k + 1hk_pos:0 < khlo:↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k !hhi:Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑k⊢ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 ≤ Real.log ↑(k + 1)! ∧
Real.log ↑(k + 1)! ≤ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 + Real.log ↑(k + 1)
k:ℕih:1 ≤ k → ↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k ! ∧ Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khn:1 ≤ k + 1hk_pos:0 < khlo:↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k !hhi:Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khfac:Real.log ↑(k + 1)! = Real.log ↑k ! + Real.log ↑(k + 1)⊢ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 ≤ Real.log ↑(k + 1)! ∧
Real.log ↑(k + 1)! ≤ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 + Real.log ↑(k + 1)
k:ℕih:1 ≤ k → ↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k ! ∧ Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khn:1 ≤ k + 1hk_pos:0 < khlo:↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k !hhi:Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khfac:Real.log ↑(k + 1)! = Real.log ↑k ! + Real.log ↑(k + 1)hstep_hi:↑(k + 1) * (Real.log ↑k - Real.log ↑(k + 1)) ≤ -1⊢ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 ≤ Real.log ↑(k + 1)! ∧
Real.log ↑(k + 1)! ≤ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 + Real.log ↑(k + 1)
k:ℕih:1 ≤ k → ↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k ! ∧ Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khn:1 ≤ k + 1hk_pos:0 < khlo:↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k !hhi:Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khfac:Real.log ↑(k + 1)! = Real.log ↑k ! + Real.log ↑(k + 1)hstep_hi:↑(k + 1) * (Real.log ↑k - Real.log ↑(k + 1)) ≤ -1hstep_lo:↑k * (Real.log ↑(k + 1) - Real.log ↑k) ≤ 1⊢ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 ≤ Real.log ↑(k + 1)! ∧
Real.log ↑(k + 1)! ≤ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 + Real.log ↑(k + 1)
k:ℕih:1 ≤ k → ↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k ! ∧ Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khn:1 ≤ k + 1hk_pos:0 < khlo:↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k !hhi:Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khfac:Real.log ↑(k + 1)! = Real.log ↑k ! + Real.log ↑(k + 1)hstep_hi:↑(k + 1) * (Real.log ↑k - Real.log ↑(k + 1)) ≤ -1hstep_lo:↑k * (Real.log ↑(k + 1) - Real.log ↑k) ≤ 1⊢ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 ≤ Real.log ↑k ! + Real.log ↑(k + 1) ∧
Real.log ↑k ! + Real.log ↑(k + 1) ≤ ↑(k + 1) * Real.log ↑(k + 1) - ↑(k + 1) + 1 + Real.log ↑(k + 1)
k:ℕih:1 ≤ k → ↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k ! ∧ Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khn:1 ≤ k + 1hk_pos:0 < khlo:↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k !hhi:Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khfac:Real.log ↑(k + 1)! = Real.log ↑k ! + Real.log (↑k + 1)hstep_lo:↑k * (Real.log (↑k + 1) - Real.log ↑k) ≤ 1hstep_hi:(↑k + 1) * (Real.log ↑k - Real.log (↑k + 1)) ≤ -1⊢ (↑k + 1) * Real.log (↑k + 1) - (↑k + 1) + 1 ≤ Real.log ↑k ! + Real.log (↑k + 1) ∧
Real.log ↑k ! + Real.log (↑k + 1) ≤ (↑k + 1) * Real.log (↑k + 1) - (↑k + 1) + 1 + Real.log (↑k + 1)
k:ℕih:1 ≤ k → ↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k ! ∧ Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khn:1 ≤ k + 1hk_pos:0 < khlo:↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k !hhi:Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khfac:Real.log ↑(k + 1)! = Real.log ↑k ! + Real.log (↑k + 1)hstep_lo:↑k * (Real.log (↑k + 1) - Real.log ↑k) ≤ 1hstep_hi:(↑k + 1) * (Real.log ↑k - Real.log (↑k + 1)) ≤ -1⊢ (↑k + 1) * Real.log (↑k + 1) - (↑k + 1) + 1 ≤ Real.log ↑k ! + Real.log (↑k + 1)k:ℕih:1 ≤ k → ↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k ! ∧ Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khn:1 ≤ k + 1hk_pos:0 < khlo:↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k !hhi:Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khfac:Real.log ↑(k + 1)! = Real.log ↑k ! + Real.log (↑k + 1)hstep_lo:↑k * (Real.log (↑k + 1) - Real.log ↑k) ≤ 1hstep_hi:(↑k + 1) * (Real.log ↑k - Real.log (↑k + 1)) ≤ -1⊢ Real.log ↑k ! + Real.log (↑k + 1) ≤ (↑k + 1) * Real.log (↑k + 1) - (↑k + 1) + 1 + Real.log (↑k + 1) k:ℕih:1 ≤ k → ↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k ! ∧ Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khn:1 ≤ k + 1hk_pos:0 < khlo:↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k !hhi:Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khfac:Real.log ↑(k + 1)! = Real.log ↑k ! + Real.log (↑k + 1)hstep_lo:↑k * (Real.log (↑k + 1) - Real.log ↑k) ≤ 1hstep_hi:(↑k + 1) * (Real.log ↑k - Real.log (↑k + 1)) ≤ -1⊢ (↑k + 1) * Real.log (↑k + 1) - (↑k + 1) + 1 ≤ Real.log ↑k ! + Real.log (↑k + 1)k:ℕih:1 ≤ k → ↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k ! ∧ Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khn:1 ≤ k + 1hk_pos:0 < khlo:↑k * Real.log ↑k - ↑k + 1 ≤ Real.log ↑k !hhi:Real.log ↑k ! ≤ ↑k * Real.log ↑k - ↑k + 1 + Real.log ↑khfac:Real.log ↑(k + 1)! = Real.log ↑k ! + Real.log (↑k + 1)hstep_lo:↑k * (Real.log (↑k + 1) - Real.log ↑k) ≤ 1hstep_hi:(↑k + 1) * (Real.log ↑k - Real.log (↑k + 1)) ≤ -1⊢ Real.log ↑k ! + Real.log (↑k + 1) ≤ (↑k + 1) * Real.log (↑k + 1) - (↑k + 1) + 1 + Real.log (↑k + 1) All goals completed! 🐙
For n ≥ 3, A_n = n! * P(n) is never a power of n.
theorem shuniaProd_not_pow (n : ℕ) (hn : 3 ≤ n) (d : ℕ) : shuniaProd n ≠ n ^ d := n:ℕhn:3 ≤ nd:ℕ⊢ shuniaProd n ≠ n ^ d
n:ℕhn:3 ≤ nd:ℕheq:shuniaProd n = n ^ d⊢ False
n:ℕhn:3 ≤ nd:ℕheq:shuniaProd n = n ^ dhdvd_fac:n - 1 ∣ n !⊢ False
n:ℕhn:3 ≤ nd:ℕheq:shuniaProd n = n ^ dhdvd_fac:n - 1 ∣ n !hdvd_prod:n - 1 ∣ shuniaProd n⊢ False
n:ℕhn:3 ≤ nd:ℕheq:shuniaProd n = n ^ dhdvd_fac:n - 1 ∣ n !hdvd_prod:n - 1 ∣ n ^ d⊢ False
n:ℕhn:3 ≤ nd:ℕheq:shuniaProd n = n ^ dhdvd_fac:n - 1 ∣ n !hdvd_prod:n - 1 ∣ n ^ dhcop:(n - 1).Coprime n⊢ False
n:ℕhn:3 ≤ nd:ℕheq:shuniaProd n = n ^ dhdvd_fac:n - 1 ∣ n !hdvd_prod:n - 1 ∣ n ^ dhcop:(n - 1).Coprime nhdvd1:n - 1 ∣ 1⊢ False
n:ℕhn:3 ≤ nd:ℕheq:shuniaProd n = n ^ dhdvd_fac:n - 1 ∣ n !hdvd_prod:n - 1 ∣ n ^ dhcop:(n - 1).Coprime nhdvd1:n - 1 ∣ 1hle1:n - 1 ≤ 1⊢ False
All goals completed! 🐙theorem offsetDigits_zero (b : ℕ) : offsetDigits b 0 = 0 := b:ℕ⊢ offsetDigits b 0 = 0
All goals completed! 🐙theorem offsetDigits_succ (b B : ℕ) :
offsetDigits b (B + 1) = if 2 ≤ b then 1 + offsetDigits b (B / b) else 0 := b:ℕB:ℕ⊢ offsetDigits b (B + 1) = if 2 ≤ b then 1 + offsetDigits b (B / b) else 0
b:ℕB:ℕ⊢ (if x : 2 ≤ b then 1 + offsetDigits b (B / b) else 0) = if 2 ≤ b then 1 + offsetDigits b (B / b) else 0; b:ℕB:ℕh✝:2 ≤ b⊢ 1 + offsetDigits b (B / b) = 1 + offsetDigits b (B / b)b:ℕB:ℕh✝:¬2 ≤ b⊢ 0 = 0 b:ℕB:ℕh✝:2 ≤ b⊢ 1 + offsetDigits b (B / b) = 1 + offsetDigits b (B / b)b:ℕB:ℕh✝:¬2 ≤ b⊢ 0 = 0 All goals completed! 🐙
For b ≥ 2 and B ≥ 1, offsetDigits b B ≥ 1.
theorem offsetDigits_pos {b B : ℕ} (hb : 2 ≤ b) (hB : 1 ≤ B) :
1 ≤ offsetDigits b B := b:ℕB:ℕhb:2 ≤ bhB:1 ≤ B⊢ 1 ≤ offsetDigits b B
b:ℕhb:2 ≤ bk:ℕhB:1 ≤ k + 1⊢ 1 ≤ offsetDigits b (k + 1)
b:ℕhb:2 ≤ bk:ℕhB:1 ≤ k + 1⊢ 1 ≤ if 2 ≤ b then 1 + offsetDigits b (k / b) else 0
b:ℕhb:2 ≤ bk:ℕhB:1 ≤ k + 1⊢ 1 ≤ 1 + offsetDigits b (k / b)
All goals completed! 🐙
Lower bound on B from offsetDigits b B: for b ≥ 2 and B ≥ 1,
b ^ (offsetDigits b B - 1) ≤ B.
theorem pow_sub_one_le_of_offsetDigits {b : ℕ} (hb : 2 ≤ b) :
∀ {B : ℕ}, 1 ≤ B → b ^ (offsetDigits b B - 1) ≤ B
b:ℕhb:2 ≤ bhB:1 ≤ 0⊢ b ^ (offsetDigits b 0 - 1) ≤ 0 b:ℕhb:2 ≤ bhB:1 ≤ 0⊢ b ^ (offsetDigits b 0 - 1) ≤ 0 All goals completed! 🐙
b:ℕhb:2 ≤ bB:ℕx✝:1 ≤ B + 1⊢ b ^ (offsetDigits b (B + 1) - 1) ≤ B + 1 b:ℕhb:2 ≤ bB:ℕx✝:1 ≤ B + 1⊢ b ^ (offsetDigits b (B + 1) - 1) ≤ B + 1
b:ℕhb:2 ≤ bB:ℕx✝:1 ≤ B + 1⊢ b ^ ((if 2 ≤ b then 1 + offsetDigits b (B / b) else 0) - 1) ≤ B + 1
b:ℕhb:2 ≤ bB:ℕx✝:1 ≤ B + 1⊢ b ^ offsetDigits b (B / b) ≤ B + 1
b:ℕhb:2 ≤ bB:ℕx✝:1 ≤ B + 1h0:0 = B / b⊢ b ^ offsetDigits b (B / b) ≤ B + 1b:ℕhb:2 ≤ bB:ℕx✝:1 ≤ B + 1hpos:0 < B / b⊢ b ^ offsetDigits b (B / b) ≤ B + 1
b:ℕhb:2 ≤ bB:ℕx✝:1 ≤ B + 1h0:0 = B / b⊢ b ^ offsetDigits b (B / b) ≤ B + 1 b:ℕhb:2 ≤ bB:ℕx✝:1 ≤ B + 1h0:0 = B / b⊢ 1 ≤ B + 1
All goals completed! 🐙
b:ℕhb:2 ≤ bB:ℕx✝:1 ≤ B + 1hpos:0 < B / b⊢ b ^ offsetDigits b (B / b) ≤ B + 1 calc b ^ offsetDigits b (B / b)
_ = b * b ^ (offsetDigits b (B / b) - 1) := b:ℕhb:2 ≤ bB:ℕx✝:1 ≤ B + 1hpos:0 < B / b⊢ b ^ offsetDigits b (B / b) = b * b ^ (offsetDigits b (B / b) - 1)
b:ℕhb:2 ≤ bB:ℕx✝:1 ≤ B + 1hpos:0 < B / bhd:1 ≤ offsetDigits b (B / b)⊢ b ^ offsetDigits b (B / b) = b * b ^ (offsetDigits b (B / b) - 1)
conv_lhs => b:ℕhb:2 ≤ bB:ℕx✝:1 ≤ B + 1hpos:0 < B / bhd:1 ≤ offsetDigits b (B / b)| b * b ^ (offsetDigits b (B / b) - 1)
_ ≤ b * (B / b) := Nat.mul_le_mul_left b (pow_sub_one_le_of_offsetDigits hb hpos)
_ ≤ B := Nat.mul_div_le B b
_ ≤ B + 1 := Nat.le_succ B
termination_by B _ => B
decreasing_by All goals completed! 🐙
Strengthened induction invariant for offsetDigits b B: for b ≥ 2 and every B : ℕ,
B + b ≤ b ^ (offsetDigits b B + 1).
lemma add_le_pow_offsetDigits_add_one {b : ℕ} (hb : 2 ≤ b) :
∀ B : ℕ, B + b ≤ b ^ (offsetDigits b B + 1)
b:ℕhb:2 ≤ b⊢ 0 + b ≤ b ^ (offsetDigits b 0 + 1) b:ℕhb:2 ≤ b⊢ 0 + b ≤ b ^ (offsetDigits b 0 + 1)
All goals completed! 🐙
b:ℕhb:2 ≤ bB:ℕ⊢ B + 1 + b ≤ b ^ (offsetDigits b (B + 1) + 1) b:ℕhb:2 ≤ bB:ℕ⊢ B + 1 + b ≤ b ^ (offsetDigits b (B + 1) + 1)
b:ℕhb:2 ≤ bB:ℕ⊢ B + 1 + b ≤ b ^ ((if 2 ≤ b then 1 + offsetDigits b (B / b) else 0) + 1)
b:ℕhb:2 ≤ bB:ℕ⊢ B + 1 + b ≤ b ^ (1 + offsetDigits b (B / b) + 1)
b:ℕhb:2 ≤ bB:ℕih:B / b + b ≤ b ^ (offsetDigits b (B / b) + 1)⊢ B + 1 + b ≤ b ^ (1 + offsetDigits b (B / b) + 1)
b:ℕhb:2 ≤ bB:ℕih:B / b + b ≤ b ^ (offsetDigits b (B / b) + 1)hmul:b * (B / b + b) ≤ b * b ^ (offsetDigits b (B / b) + 1)⊢ B + 1 + b ≤ b ^ (1 + offsetDigits b (B / b) + 1)
b:ℕhb:2 ≤ bB:ℕih:B / b + b ≤ b ^ (offsetDigits b (B / b) + 1)hmul:b * (B / b + b) ≤ b * b ^ (offsetDigits b (B / b) + 1)hdiv:B + 1 ≤ b * (B / b) + b⊢ B + 1 + b ≤ b ^ (1 + offsetDigits b (B / b) + 1)
b:ℕhb:2 ≤ bB:ℕih:B / b + b ≤ b ^ (offsetDigits b (B / b) + 1)hmul:b * (B / b + b) ≤ b * b ^ (offsetDigits b (B / b) + 1)hdiv:B + 1 ≤ b * (B / b) + bhpow:b ^ (1 + offsetDigits b (B / b) + 1) = b * b ^ (offsetDigits b (B / b) + 1)⊢ B + 1 + b ≤ b ^ (1 + offsetDigits b (B / b) + 1)
b:ℕhb:2 ≤ bB:ℕih:B / b + b ≤ b ^ (offsetDigits b (B / b) + 1)hmul:b * (B / b + b) ≤ b * b ^ (offsetDigits b (B / b) + 1)hdiv:B + 1 ≤ b * (B / b) + bhpow:b ^ (1 + offsetDigits b (B / b) + 1) = b * b ^ (offsetDigits b (B / b) + 1)⊢ B + 1 + b ≤ b * b ^ (offsetDigits b (B / b) + 1)
All goals completed! 🐙
termination_by B => B
decreasing_by All goals completed! 🐙
Upper bound on B from offsetDigits b B: for b ≥ 2 and every B : ℕ,
B < b ^ (offsetDigits b B + 1).
theorem offsetDigits_pow_gt {b : ℕ} (hb : 2 ≤ b) (B : ℕ) :
B < b ^ (offsetDigits b B + 1) := b:ℕhb:2 ≤ bB:ℕ⊢ B < b ^ (offsetDigits b B + 1)
b:ℕhb:2 ≤ bB:ℕh:B + b ≤ b ^ (offsetDigits b B + 1)⊢ B < b ^ (offsetDigits b B + 1)
All goals completed! 🐙
Forward bridge inequality (shuniaDeficit to θ):
for every n ≥ 2 with 1 ≤ shuniaDeficit n, (shuniaDeficit n : ℝ) * log n ≤ n - θ(n) - 1.
theorem shuniaDeficit_mul_log_le {n : ℕ} (hn : 2 ≤ n) (hpos : 1 ≤ shuniaDeficit n) :
(shuniaDeficit n : ℝ) * Real.log n ≤ (n : ℝ) - Chebyshev.theta n - 1 := n:ℕhn:2 ≤ nhpos:1 ≤ shuniaDeficit n⊢ ↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1
n:ℕhn:2 ≤ nhpos:1 ≤ shuniaDeficit nhprod_le:shuniaProd n ≤ n ^ (offsetDigits n (shuniaProd n - 1) + 1)⊢ ↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1
n:ℕhn:2 ≤ nhpos:1 ≤ shuniaDeficit nd:ℕ := offsetDigits n (shuniaProd n - 1)hprod_le:shuniaProd n ≤ n ^ (d + 1)⊢ ↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1
n:ℕhn:2 ≤ nhpos:1 ≤ shuniaDeficit nd:ℕ := offsetDigits n (shuniaProd n - 1)hprod_le:shuniaProd n ≤ n ^ (d + 1)hlog_le:Real.log ↑(shuniaProd n) ≤ ↑(d + 1) * Real.log ↑n⊢ ↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1
n:ℕhn:2 ≤ nhpos:1 ≤ shuniaDeficit nd:ℕ := offsetDigits n (shuniaProd n - 1)hprod_le:shuniaProd n ≤ n ^ (d + 1)hlog_le:Real.log ↑(shuniaProd n) ≤ ↑(d + 1) * Real.log ↑nhlog_pos:0 < Real.log ↑n⊢ ↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1
n:ℕhn:2 ≤ nhpos:1 ≤ shuniaDeficit nd:ℕ := offsetDigits n (shuniaProd n - 1)hprod_le:shuniaProd n ≤ n ^ (d + 1)hlog_le:Real.log ↑(shuniaProd n) ≤ ↑(d + 1) * Real.log ↑nhlog_pos:0 < Real.log ↑nhfac_lo:↑n * Real.log ↑n - ↑n + 1 ≤ Real.log ↑n !⊢ ↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1
n:ℕhn:2 ≤ nhpos:1 ≤ shuniaDeficit nd:ℕ := offsetDigits n (shuniaProd n - 1)hprod_le:shuniaProd n ≤ n ^ (d + 1)hlog_le:Real.log ↑(shuniaProd n) ≤ ↑(d + 1) * Real.log ↑nhlog_pos:0 < Real.log ↑nhfac_lo:↑n * Real.log ↑n - ↑n + 1 ≤ Real.log ↑n !hlog_eq:Real.log ↑(shuniaProd n) = Real.log ↑n ! + Chebyshev.theta ↑n⊢ ↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1
n:ℕhn:2 ≤ nhpos:1 ≤ shuniaDeficit nd:ℕ := offsetDigits n (shuniaProd n - 1)hprod_le:shuniaProd n ≤ n ^ (d + 1)hlog_le:Real.log ↑(shuniaProd n) ≤ ↑(d + 1) * Real.log ↑nhlog_pos:0 < Real.log ↑nhfac_lo:↑n * Real.log ↑n - ↑n + 1 ≤ Real.log ↑n !hlog_eq:Real.log ↑(shuniaProd n) = Real.log ↑n ! + Chebyshev.theta ↑nhd_le:d ≤ n - 1⊢ ↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1
n:ℕhn:2 ≤ nhpos:1 ≤ shuniaDeficit nd:ℕ := offsetDigits n (shuniaProd n - 1)hprod_le:shuniaProd n ≤ n ^ (d + 1)hlog_le:Real.log ↑(shuniaProd n) ≤ ↑(d + 1) * Real.log ↑nhlog_pos:0 < Real.log ↑nhfac_lo:↑n * Real.log ↑n - ↑n + 1 ≤ Real.log ↑n !hlog_eq:Real.log ↑(shuniaProd n) = Real.log ↑n ! + Chebyshev.theta ↑nhd_le:d ≤ n - 1⊢ ↑(n - 1 - offsetDigits n (shuniaProd n - 1)) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1
n:ℕhn:2 ≤ nhpos:1 ≤ shuniaDeficit nd:ℕ := offsetDigits n (shuniaProd n - 1)hprod_le:shuniaProd n ≤ n ^ (d + 1)hlog_le:Real.log ↑(shuniaProd n) ≤ ↑(d + 1) * Real.log ↑nhlog_pos:0 < Real.log ↑nhfac_lo:↑n * Real.log ↑n - ↑n + 1 ≤ Real.log ↑n !hlog_eq:Real.log ↑(shuniaProd n) = Real.log ↑n ! + Chebyshev.theta ↑nhd_le:d ≤ n - 1hcast:↑(n - 1 - d) = ↑n - 1 - ↑d⊢ ↑(n - 1 - offsetDigits n (shuniaProd n - 1)) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1
n:ℕhn:2 ≤ nhpos:1 ≤ shuniaDeficit nd:ℕ := offsetDigits n (shuniaProd n - 1)hprod_le:shuniaProd n ≤ n ^ (d + 1)hlog_le:Real.log ↑(shuniaProd n) ≤ ↑(d + 1) * Real.log ↑nhlog_pos:0 < Real.log ↑nhfac_lo:↑n * Real.log ↑n - ↑n + 1 ≤ Real.log ↑n !hlog_eq:Real.log ↑(shuniaProd n) = Real.log ↑n ! + Chebyshev.theta ↑nhd_le:d ≤ n - 1hcast:↑(n - 1 - d) = ↑n - 1 - ↑d⊢ (↑n - 1 - ↑d) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1
n:ℕhn:2 ≤ nhpos:1 ≤ shuniaDeficit nd:ℕ := offsetDigits n (shuniaProd n - 1)hprod_le:shuniaProd n ≤ n ^ (d + 1)hlog_pos:0 < Real.log ↑nhfac_lo:↑n * Real.log ↑n - ↑n + 1 ≤ Real.log ↑n !hlog_eq:Real.log ↑(shuniaProd n) = Real.log ↑n ! + Chebyshev.theta ↑nhd_le:d ≤ n - 1hcast:↑(n - 1 - d) = ↑n - 1 - ↑dhlog_le:Real.log ↑(shuniaProd n) ≤ (↑d + 1) * Real.log ↑n⊢ (↑n - 1 - ↑d) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1
All goals completed! 🐙
Reverse bridge inequality (θ to shuniaDeficit):
for every n ≥ 2, (n : ℝ) - θ(n) ≤ (shuniaDeficit n + 3 : ℝ) * log n + 1.
theorem self_sub_theta_le_shuniaDeficit_mul_log {n : ℕ} (hn : 2 ≤ n) :
(n : ℝ) - Chebyshev.theta n ≤ (shuniaDeficit n + 3 : ℝ) * Real.log n + 1 := n:ℕhn:2 ≤ n⊢ ↑n - Chebyshev.theta ↑n ≤ (↑(shuniaDeficit n) + 3) * Real.log ↑n + 1
n:ℕhn:2 ≤ nd:ℕ := offsetDigits n (shuniaProd n - 1)⊢ ↑n - Chebyshev.theta ↑n ≤ (↑(shuniaDeficit n) + 3) * Real.log ↑n + 1
n:ℕhn:2 ≤ nd:ℕ := offsetDigits n (shuniaProd n - 1)h4:4 ≤ shuniaProd n⊢ ↑n - Chebyshev.theta ↑n ≤ (↑(shuniaDeficit n) + 3) * Real.log ↑n + 1
n:ℕhn:2 ≤ nd:ℕ := offsetDigits n (shuniaProd n - 1)h4:4 ≤ shuniaProd nhB1:1 ≤ shuniaProd n - 1⊢ ↑n - Chebyshev.theta ↑n ≤ (↑(shuniaDeficit n) + 3) * Real.log ↑n + 1
n:ℕhn:2 ≤ nd:ℕ := offsetDigits n (shuniaProd n - 1)h4:4 ≤ shuniaProd nhB1:1 ≤ shuniaProd n - 1hd1:1 ≤ d⊢ ↑n - Chebyshev.theta ↑n ≤ (↑(shuniaDeficit n) + 3) * Real.log ↑n + 1
n:ℕhn:2 ≤ nd:ℕ := offsetDigits n (shuniaProd n - 1)h4:4 ≤ shuniaProd nhB1:1 ≤ shuniaProd n - 1hd1:1 ≤ dhpow_le:n ^ (d - 1) ≤ shuniaProd n⊢ ↑n - Chebyshev.theta ↑n ≤ (↑(shuniaDeficit n) + 3) * Real.log ↑n + 1
n:ℕhn:2 ≤ nd:ℕ := offsetDigits n (shuniaProd n - 1)h4:4 ≤ shuniaProd nhB1:1 ≤ shuniaProd n - 1hd1:1 ≤ dhpow_le:n ^ (d - 1) ≤ shuniaProd nhlog_lo:↑(d - 1) * Real.log ↑n ≤ Real.log ↑(shuniaProd n)⊢ ↑n - Chebyshev.theta ↑n ≤ (↑(shuniaDeficit n) + 3) * Real.log ↑n + 1
n:ℕhn:2 ≤ nd:ℕ := offsetDigits n (shuniaProd n - 1)h4:4 ≤ shuniaProd nhB1:1 ≤ shuniaProd n - 1hd1:1 ≤ dhpow_le:n ^ (d - 1) ≤ shuniaProd nhlog_lo:↑(d - 1) * Real.log ↑n ≤ Real.log ↑(shuniaProd n)hfac_hi:Real.log ↑n ! ≤ ↑n * Real.log ↑n - ↑n + 1 + Real.log ↑n⊢ ↑n - Chebyshev.theta ↑n ≤ (↑(shuniaDeficit n) + 3) * Real.log ↑n + 1
n:ℕhn:2 ≤ nd:ℕ := offsetDigits n (shuniaProd n - 1)h4:4 ≤ shuniaProd nhB1:1 ≤ shuniaProd n - 1hd1:1 ≤ dhpow_le:n ^ (d - 1) ≤ shuniaProd nhlog_lo:↑(d - 1) * Real.log ↑n ≤ Real.log ↑(shuniaProd n)hfac_hi:Real.log ↑n ! ≤ ↑n * Real.log ↑n - ↑n + 1 + Real.log ↑nhlog_eq:Real.log ↑(shuniaProd n) = Real.log ↑n ! + Chebyshev.theta ↑n⊢ ↑n - Chebyshev.theta ↑n ≤ (↑(shuniaDeficit n) + 3) * Real.log ↑n + 1
n:ℕhn:2 ≤ nd:ℕ := offsetDigits n (shuniaProd n - 1)h4:4 ≤ shuniaProd nhB1:1 ≤ shuniaProd n - 1hd1:1 ≤ dhpow_le:n ^ (d - 1) ≤ shuniaProd nhlog_lo:↑(d - 1) * Real.log ↑n ≤ Real.log ↑(shuniaProd n)hfac_hi:Real.log ↑n ! ≤ ↑n * Real.log ↑n - ↑n + 1 + Real.log ↑nhlog_eq:Real.log ↑(shuniaProd n) = Real.log ↑n ! + Chebyshev.theta ↑nhlog_pos:0 < Real.log ↑n⊢ ↑n - Chebyshev.theta ↑n ≤ (↑(shuniaDeficit n) + 3) * Real.log ↑n + 1
n:ℕhn:2 ≤ nd:ℕ := offsetDigits n (shuniaProd n - 1)h4:4 ≤ shuniaProd nhB1:1 ≤ shuniaProd n - 1hd1:1 ≤ dhpow_le:n ^ (d - 1) ≤ shuniaProd nhlog_lo:↑(d - 1) * Real.log ↑n ≤ Real.log ↑(shuniaProd n)hfac_hi:Real.log ↑n ! ≤ ↑n * Real.log ↑n - ↑n + 1 + Real.log ↑nhlog_eq:Real.log ↑(shuniaProd n) = Real.log ↑n ! + Chebyshev.theta ↑nhlog_pos:0 < Real.log ↑nhd_cast:↑(d - 1) = ↑d - 1⊢ ↑n - Chebyshev.theta ↑n ≤ (↑(shuniaDeficit n) + 3) * Real.log ↑n + 1
n:ℕhn:2 ≤ nd:ℕ := offsetDigits n (shuniaProd n - 1)h4:4 ≤ shuniaProd nhB1:1 ≤ shuniaProd n - 1hd1:1 ≤ dhpow_le:n ^ (d - 1) ≤ shuniaProd nhlog_lo:(↑d - 1) * Real.log ↑n ≤ Real.log ↑(shuniaProd n)hfac_hi:Real.log ↑n ! ≤ ↑n * Real.log ↑n - ↑n + 1 + Real.log ↑nhlog_eq:Real.log ↑(shuniaProd n) = Real.log ↑n ! + Chebyshev.theta ↑nhlog_pos:0 < Real.log ↑nhd_cast:↑(d - 1) = ↑d - 1⊢ ↑n - Chebyshev.theta ↑n ≤ (↑(shuniaDeficit n) + 3) * Real.log ↑n + 1
n:ℕhn:2 ≤ nd:ℕ := offsetDigits n (shuniaProd n - 1)h4:4 ≤ shuniaProd nhB1:1 ≤ shuniaProd n - 1hd1:1 ≤ dhpow_le:n ^ (d - 1) ≤ shuniaProd nhlog_lo:(↑d - 1) * Real.log ↑n ≤ Real.log ↑(shuniaProd n)hfac_hi:Real.log ↑n ! ≤ ↑n * Real.log ↑n - ↑n + 1 + Real.log ↑nhlog_eq:Real.log ↑(shuniaProd n) = Real.log ↑n ! + Chebyshev.theta ↑nhlog_pos:0 < Real.log ↑nhd_cast:↑(d - 1) = ↑d - 1hdef_ge:↑n - 1 - ↑d ≤ ↑(shuniaDeficit n)⊢ ↑n - Chebyshev.theta ↑n ≤ (↑(shuniaDeficit n) + 3) * Real.log ↑n + 1
All goals completed! 🐙
3. Quartering Clock & Unconditional Verification for 2 ≤ n ≤ 21845
lemma quarterClock_bounds_aux :
∀ z : ℕ, 4 ^ quarterClock z ≤ 3 * z + 1 ∧ 3 * z + 3 < 4 ^ (quarterClock z + 1)
⊢ 4 ^ quarterClock 0 ≤ 3 * 0 + 1 ∧ 3 * 0 + 3 < 4 ^ (quarterClock 0 + 1) ⊢ 4 ^ quarterClock 0 ≤ 3 * 0 + 1 ∧ 3 * 0 + 3 < 4 ^ (quarterClock 0 + 1) All goals completed! 🐙
z:ℕ⊢ 4 ^ quarterClock (z + 1) ≤ 3 * (z + 1) + 1 ∧ 3 * (z + 1) + 3 < 4 ^ (quarterClock (z + 1) + 1) z:ℕ⊢ 4 ^ quarterClock (z + 1) ≤ 3 * (z + 1) + 1 ∧ 3 * (z + 1) + 3 < 4 ^ (quarterClock (z + 1) + 1)
z:ℕ⊢ 4 ^ (1 + quarterClock (z / 4)) ≤ 3 * (z + 1) + 1 ∧ 3 * (z + 1) + 3 < 4 ^ (1 + quarterClock (z / 4) + 1)
z:ℕhlo:4 ^ quarterClock (z / 4) ≤ 3 * (z / 4) + 1hhi:3 * (z / 4) + 3 < 4 ^ (quarterClock (z / 4) + 1)⊢ 4 ^ (1 + quarterClock (z / 4)) ≤ 3 * (z + 1) + 1 ∧ 3 * (z + 1) + 3 < 4 ^ (1 + quarterClock (z / 4) + 1)
z:ℕhlo:4 ^ quarterClock (z / 4) ≤ 3 * (z / 4) + 1hhi:3 * (z / 4) + 3 < 4 ^ (quarterClock (z / 4) + 1)hdiv1:4 * (z / 4) ≤ z⊢ 4 ^ (1 + quarterClock (z / 4)) ≤ 3 * (z + 1) + 1 ∧ 3 * (z + 1) + 3 < 4 ^ (1 + quarterClock (z / 4) + 1)
z:ℕhlo:4 ^ quarterClock (z / 4) ≤ 3 * (z / 4) + 1hhi:3 * (z / 4) + 3 < 4 ^ (quarterClock (z / 4) + 1)hdiv1:4 * (z / 4) ≤ zhdiv2:z < 4 * (z / 4) + 4⊢ 4 ^ (1 + quarterClock (z / 4)) ≤ 3 * (z + 1) + 1 ∧ 3 * (z + 1) + 3 < 4 ^ (1 + quarterClock (z / 4) + 1)
z:ℕhlo:4 ^ quarterClock (z / 4) ≤ 3 * (z / 4) + 1hhi:3 * (z / 4) + 3 < 4 ^ (quarterClock (z / 4) + 1)hdiv1:4 * (z / 4) ≤ zhdiv2:z < 4 * (z / 4) + 4hpow1:4 ^ (1 + quarterClock (z / 4)) = 4 * 4 ^ quarterClock (z / 4)⊢ 4 ^ (1 + quarterClock (z / 4)) ≤ 3 * (z + 1) + 1 ∧ 3 * (z + 1) + 3 < 4 ^ (1 + quarterClock (z / 4) + 1)
z:ℕhlo:4 ^ quarterClock (z / 4) ≤ 3 * (z / 4) + 1hhi:3 * (z / 4) + 3 < 4 ^ (quarterClock (z / 4) + 1)hdiv1:4 * (z / 4) ≤ zhdiv2:z < 4 * (z / 4) + 4hpow1:4 ^ (1 + quarterClock (z / 4)) = 4 * 4 ^ quarterClock (z / 4)hpow2:4 ^ (1 + quarterClock (z / 4) + 1) = 4 * 4 ^ (quarterClock (z / 4) + 1)⊢ 4 ^ (1 + quarterClock (z / 4)) ≤ 3 * (z + 1) + 1 ∧ 3 * (z + 1) + 3 < 4 ^ (1 + quarterClock (z / 4) + 1)
z:ℕhlo:4 ^ quarterClock (z / 4) ≤ 3 * (z / 4) + 1hhi:3 * (z / 4) + 3 < 4 ^ (quarterClock (z / 4) + 1)hdiv1:4 * (z / 4) ≤ zhdiv2:z < 4 * (z / 4) + 4hpow1:4 ^ (1 + quarterClock (z / 4)) = 4 * 4 ^ quarterClock (z / 4)hpow2:4 ^ (1 + quarterClock (z / 4) + 1) = 4 * 4 ^ (quarterClock (z / 4) + 1)⊢ 4 * 4 ^ quarterClock (z / 4) ≤ 3 * (z + 1) + 1 ∧ 3 * (z + 1) + 3 < 4 * 4 ^ (quarterClock (z / 4) + 1)
z:ℕhlo:4 ^ quarterClock (z / 4) ≤ 3 * (z / 4) + 1hhi:3 * (z / 4) + 3 < 4 ^ (quarterClock (z / 4) + 1)hdiv1:4 * (z / 4) ≤ zhdiv2:z < 4 * (z / 4) + 4hpow1:4 ^ (1 + quarterClock (z / 4)) = 4 * 4 ^ quarterClock (z / 4)hpow2:4 ^ (1 + quarterClock (z / 4) + 1) = 4 * 4 ^ (quarterClock (z / 4) + 1)⊢ 4 * 4 ^ quarterClock (z / 4) ≤ 3 * (z + 1) + 1z:ℕhlo:4 ^ quarterClock (z / 4) ≤ 3 * (z / 4) + 1hhi:3 * (z / 4) + 3 < 4 ^ (quarterClock (z / 4) + 1)hdiv1:4 * (z / 4) ≤ zhdiv2:z < 4 * (z / 4) + 4hpow1:4 ^ (1 + quarterClock (z / 4)) = 4 * 4 ^ quarterClock (z / 4)hpow2:4 ^ (1 + quarterClock (z / 4) + 1) = 4 * 4 ^ (quarterClock (z / 4) + 1)⊢ 3 * (z + 1) + 3 < 4 * 4 ^ (quarterClock (z / 4) + 1) z:ℕhlo:4 ^ quarterClock (z / 4) ≤ 3 * (z / 4) + 1hhi:3 * (z / 4) + 3 < 4 ^ (quarterClock (z / 4) + 1)hdiv1:4 * (z / 4) ≤ zhdiv2:z < 4 * (z / 4) + 4hpow1:4 ^ (1 + quarterClock (z / 4)) = 4 * 4 ^ quarterClock (z / 4)hpow2:4 ^ (1 + quarterClock (z / 4) + 1) = 4 * 4 ^ (quarterClock (z / 4) + 1)⊢ 4 * 4 ^ quarterClock (z / 4) ≤ 3 * (z + 1) + 1z:ℕhlo:4 ^ quarterClock (z / 4) ≤ 3 * (z / 4) + 1hhi:3 * (z / 4) + 3 < 4 ^ (quarterClock (z / 4) + 1)hdiv1:4 * (z / 4) ≤ zhdiv2:z < 4 * (z / 4) + 4hpow1:4 ^ (1 + quarterClock (z / 4)) = 4 * 4 ^ quarterClock (z / 4)hpow2:4 ^ (1 + quarterClock (z / 4) + 1) = 4 * 4 ^ (quarterClock (z / 4) + 1)⊢ 3 * (z + 1) + 3 < 4 * 4 ^ (quarterClock (z / 4) + 1) All goals completed! 🐙
Specification of the quartering clock:
4 ^ quarterClock (n - 1) ≤ 3 * n < 4 ^ (quarterClock (n - 1) + 1) for all n ≥ 1.
theorem quarterClock_spec (n : ℕ) (hn : 1 ≤ n) :
4 ^ quarterClock (n - 1) ≤ 3 * n ∧ 3 * n < 4 ^ (quarterClock (n - 1) + 1) := n:ℕhn:1 ≤ n⊢ 4 ^ quarterClock (n - 1) ≤ 3 * n ∧ 3 * n < 4 ^ (quarterClock (n - 1) + 1)
n:ℕhn:1 ≤ nhlo:4 ^ quarterClock (n - 1) ≤ 3 * (n - 1) + 1hhi:3 * (n - 1) + 3 < 4 ^ (quarterClock (n - 1) + 1)⊢ 4 ^ quarterClock (n - 1) ≤ 3 * n ∧ 3 * n < 4 ^ (quarterClock (n - 1) + 1)
exact ⟨n:ℕhn:1 ≤ nhlo:4 ^ quarterClock (n - 1) ≤ 3 * (n - 1) + 1hhi:3 * (n - 1) + 3 < 4 ^ (quarterClock (n - 1) + 1)⊢ 4 ^ quarterClock (n - 1) ≤ 3 * n All goals completed! 🐙, n:ℕhn:1 ≤ nhlo:4 ^ quarterClock (n - 1) ≤ 3 * (n - 1) + 1hhi:3 * (n - 1) + 3 < 4 ^ (quarterClock (n - 1) + 1)⊢ 3 * n < 4 ^ (quarterClock (n - 1) + 1) All goals completed! 🐙⟩
Uniqueness of quarterClock (n - 1) from the 4 ^ q ≤ 3 * n < 4 ^ (q + 1) sandwich.
theorem quarterClock_eq_of_bounds {n q : ℕ} (hn : 1 ≤ n)
(hlo : 4 ^ q ≤ 3 * n) (hhi : 3 * n < 4 ^ (q + 1)) :
quarterClock (n - 1) = q := n:ℕq:ℕhn:1 ≤ nhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)⊢ quarterClock (n - 1) = q
n:ℕq:ℕhn:1 ≤ nhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hlo':4 ^ quarterClock (n - 1) ≤ 3 * nhhi':3 * n < 4 ^ (quarterClock (n - 1) + 1)⊢ quarterClock (n - 1) = q
n:ℕq:ℕhn:1 ≤ nhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hlo':4 ^ quarterClock (n - 1) ≤ 3 * nhhi':3 * n < 4 ^ (quarterClock (n - 1) + 1)hne:¬quarterClock (n - 1) = q⊢ False
n:ℕq:ℕhn:1 ≤ nhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hlo':4 ^ quarterClock (n - 1) ≤ 3 * nhhi':3 * n < 4 ^ (quarterClock (n - 1) + 1)hne:¬quarterClock (n - 1) = qhlt:quarterClock (n - 1) < q⊢ Falsen:ℕq:ℕhn:1 ≤ nhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hlo':4 ^ quarterClock (n - 1) ≤ 3 * nhhi':3 * n < 4 ^ (quarterClock (n - 1) + 1)hne:¬quarterClock (n - 1) = qhgt:q < quarterClock (n - 1)⊢ False
n:ℕq:ℕhn:1 ≤ nhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hlo':4 ^ quarterClock (n - 1) ≤ 3 * nhhi':3 * n < 4 ^ (quarterClock (n - 1) + 1)hne:¬quarterClock (n - 1) = qhlt:quarterClock (n - 1) < q⊢ False n:ℕq:ℕhn:1 ≤ nhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hlo':4 ^ quarterClock (n - 1) ≤ 3 * nhhi':3 * n < 4 ^ (quarterClock (n - 1) + 1)hne:¬quarterClock (n - 1) = qhlt:quarterClock (n - 1) < qhpow:4 ^ (quarterClock (n - 1) + 1) ≤ 4 ^ q⊢ False
All goals completed! 🐙
n:ℕq:ℕhn:1 ≤ nhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hlo':4 ^ quarterClock (n - 1) ≤ 3 * nhhi':3 * n < 4 ^ (quarterClock (n - 1) + 1)hne:¬quarterClock (n - 1) = qhgt:q < quarterClock (n - 1)⊢ False n:ℕq:ℕhn:1 ≤ nhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hlo':4 ^ quarterClock (n - 1) ≤ 3 * nhhi':3 * n < 4 ^ (quarterClock (n - 1) + 1)hne:¬quarterClock (n - 1) = qhgt:q < quarterClock (n - 1)hpow:4 ^ (q + 1) ≤ 4 ^ quarterClock (n - 1)⊢ False
All goals completed! 🐙
General dyadic helper for 2 ≤ n ≤ 1365:
if a ≤ n ≤ b, 4 ^ q ≤ 3 * n < 4 ^ (q + 1), 2 ^ m ≤ a ^ p (p ≥ 1), and
p * (b - 1) < (q + 1) * 2 ^ q * m * (693147 / 1000000), then ¬ ShuniaCounterexample n.
lemma not_shuniaCounterexample_of_dyadic_small {n a b q m p : ℕ}
(hn2 : 2 ≤ n) (ha : a ≤ n) (hb : n ≤ b) (hp : 1 ≤ p)
(hlo : 4 ^ q ≤ 3 * n) (hhi : 3 * n < 4 ^ (q + 1))
(hpow : 2 ^ m ≤ a ^ p)
(hineq : (p : ℝ) * (b - 1 : ℝ) <
((q + 1) * 2 ^ q * m : ℝ) * (693147 / 1000000 : ℝ)) :
¬ ShuniaCounterexample n := n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)⊢ ¬ShuniaCounterexample n
n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)⊢ ¬(quarterClock (n - 1) + 1) * 2 ^ quarterClock (n - 1) ≤ shuniaDeficit n
n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)⊢ ¬(q + 1) * 2 ^ q ≤ shuniaDeficit n
n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit n⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑n⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑n ^ p⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑n ^ phlog_m:↑m * (693147 / 1000000) ≤ ↑p * Real.log ↑n⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑n ^ phlog_m:↑m * (693147 / 1000000) ≤ ↑p * Real.log ↑nhdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑n ^ phlog_m:↑m * (693147 / 1000000) ≤ ↑p * Real.log ↑nhdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1htheta:0 ≤ Chebyshev.theta ↑n⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑n ^ phlog_m:↑m * (693147 / 1000000) ≤ ↑p * Real.log ↑nhdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1htheta:0 ≤ Chebyshev.theta ↑nhce_r:↑((q + 1) * 2 ^ q) * Real.log ↑n ≤ ↑n - 1⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑n ^ phlog_m:↑m * (693147 / 1000000) ≤ ↑p * Real.log ↑nhdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1htheta:0 ≤ Chebyshev.theta ↑nhce_r:↑((q + 1) * 2 ^ q) * Real.log ↑n ≤ ↑n - 1hmul1:↑((q + 1) * 2 ^ q) * (↑m * (693147 / 1000000)) ≤ ↑((q + 1) * 2 ^ q) * (↑p * Real.log ↑n)⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑n ^ phlog_m:↑m * (693147 / 1000000) ≤ ↑p * Real.log ↑nhdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1htheta:0 ≤ Chebyshev.theta ↑nhce_r:↑((q + 1) * 2 ^ q) * Real.log ↑n ≤ ↑n - 1hmul1:↑((q + 1) * 2 ^ q) * (↑m * (693147 / 1000000)) ≤ ↑((q + 1) * 2 ^ q) * (↑p * Real.log ↑n)hmul2:↑p * (↑((q + 1) * 2 ^ q) * Real.log ↑n) ≤ ↑p * (↑n - 1)⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑n ^ phlog_m:↑m * (693147 / 1000000) ≤ ↑p * Real.log ↑nhdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1htheta:0 ≤ Chebyshev.theta ↑nhce_r:↑((q + 1) * 2 ^ q) * Real.log ↑n ≤ ↑n - 1hmul1:↑((q + 1) * 2 ^ q) * (↑m * (693147 / 1000000)) ≤ ↑((q + 1) * 2 ^ q) * (↑p * Real.log ↑n)hmul2:↑p * (↑((q + 1) * 2 ^ q) * Real.log ↑n) ≤ ↑p * (↑n - 1)hn_le_b:↑n - 1 ≤ ↑b - 1⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑n ^ phlog_m:↑m * (693147 / 1000000) ≤ ↑p * Real.log ↑nhdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1htheta:0 ≤ Chebyshev.theta ↑nhce_r:↑((q + 1) * 2 ^ q) * Real.log ↑n ≤ ↑n - 1hmul1:↑((q + 1) * 2 ^ q) * (↑m * (693147 / 1000000)) ≤ ↑((q + 1) * 2 ^ q) * (↑p * Real.log ↑n)hmul2:↑p * (↑((q + 1) * 2 ^ q) * Real.log ↑n) ≤ ↑p * (↑n - 1)hn_le_b:↑n - 1 ≤ ↑b - 1hmul3:↑p * (↑n - 1) ≤ ↑p * (↑b - 1)⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕp:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhp:1 ≤ phlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ a ^ phineq:↑p * (↑b - 1) < (↑q + 1) * 2 ^ q * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑n ^ phlog_m:↑m * (693147 / 1000000) ≤ ↑p * Real.log ↑nhdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1htheta:0 ≤ Chebyshev.theta ↑nhce_r:↑((q + 1) * 2 ^ q) * Real.log ↑n ≤ ↑n - 1hn_le_b:↑n - 1 ≤ ↑b - 1hmul3:↑p * (↑n - 1) ≤ ↑p * (↑b - 1)hmul1:(↑q + 1) * 2 ^ q * (↑m * (693147 / 1000000)) ≤ (↑q + 1) * 2 ^ q * (↑p * Real.log ↑n)hmul2:↑p * ((↑q + 1) * 2 ^ q * Real.log ↑n) ≤ ↑p * (↑n - 1)⊢ False
All goals completed! 🐙
Unconditional verification for 2 ≤ n ≤ 1365 using only factorial centering.
theorem not_shuniaCounterexample_of_le_1365 {n : ℕ} (hn2 : 2 ≤ n) (hn : n ≤ 1365) :
¬ ShuniaCounterexample n := n:ℕhn2:2 ≤ nhn:n ≤ 1365⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:n ≤ 3⊢ ¬ShuniaCounterexample nn:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:n ≤ 3⊢ ¬ShuniaCounterexample n exact not_shuniaCounterexample_of_dyadic_small hn2 (a := 2) (b := 3) (q := 1)
(m := 1) (p := 1) hn2 h3 (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:n ≤ 3⊢ 1 ≤ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:n ≤ 3⊢ 4 ^ 1 ≤ 3 * n All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:n ≤ 3⊢ 3 * n < 4 ^ (1 + 1) All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:n ≤ 3⊢ 2 ^ 1 ≤ 2 ^ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:n ≤ 3⊢ ↑1 * (↑3 - 1) < (↑1 + 1) * 2 ^ 1 * ↑1 * (693147 / 1000000) All goals completed! 🐙)
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:n ≤ 5⊢ ¬ShuniaCounterexample nn:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:n ≤ 5⊢ ¬ShuniaCounterexample n exact not_shuniaCounterexample_of_dyadic_small hn2 (a := 4) (b := 5) (q := 1)
(m := 2) (p := 1) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:n ≤ 5⊢ 4 ≤ n All goals completed! 🐙) h5 (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:n ≤ 5⊢ 1 ≤ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:n ≤ 5⊢ 4 ^ 1 ≤ 3 * n All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:n ≤ 5⊢ 3 * n < 4 ^ (1 + 1) All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:n ≤ 5⊢ 2 ^ 2 ≤ 4 ^ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:n ≤ 5⊢ ↑1 * (↑5 - 1) < (↑1 + 1) * 2 ^ 1 * ↑2 * (693147 / 1000000) All goals completed! 🐙)
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:n ≤ 17⊢ ¬ShuniaCounterexample nn:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:n ≤ 17⊢ ¬ShuniaCounterexample n exact not_shuniaCounterexample_of_dyadic_small hn2 (a := 6) (b := 17) (q := 2)
(m := 2) (p := 1) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:n ≤ 17⊢ 6 ≤ n All goals completed! 🐙) h17 (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:n ≤ 17⊢ 1 ≤ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:n ≤ 17⊢ 4 ^ 2 ≤ 3 * n All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:n ≤ 17⊢ 3 * n < 4 ^ (2 + 1) All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:n ≤ 17⊢ 2 ^ 2 ≤ 6 ^ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:n ≤ 17⊢ ↑1 * (↑17 - 1) < (↑2 + 1) * 2 ^ 2 * ↑2 * (693147 / 1000000) All goals completed! 🐙)
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:n ≤ 21⊢ ¬ShuniaCounterexample nn:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:n ≤ 21⊢ ¬ShuniaCounterexample n exact not_shuniaCounterexample_of_dyadic_small hn2 (a := 18) (b := 21) (q := 2)
(m := 4) (p := 1) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:n ≤ 21⊢ 18 ≤ n All goals completed! 🐙) h21 (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:n ≤ 21⊢ 1 ≤ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:n ≤ 21⊢ 4 ^ 2 ≤ 3 * n All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:n ≤ 21⊢ 3 * n < 4 ^ (2 + 1) All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:n ≤ 21⊢ 2 ^ 4 ≤ 18 ^ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:n ≤ 21⊢ ↑1 * (↑21 - 1) < (↑2 + 1) * 2 ^ 2 * ↑4 * (693147 / 1000000) All goals completed! 🐙)
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:n ≤ 85⊢ ¬ShuniaCounterexample nn:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:n ≤ 85⊢ ¬ShuniaCounterexample n exact not_shuniaCounterexample_of_dyadic_small hn2 (a := 22) (b := 85) (q := 3)
(m := 4) (p := 1) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:n ≤ 85⊢ 22 ≤ n All goals completed! 🐙) h85 (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:n ≤ 85⊢ 1 ≤ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:n ≤ 85⊢ 4 ^ 3 ≤ 3 * n All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:n ≤ 85⊢ 3 * n < 4 ^ (3 + 1) All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:n ≤ 85⊢ 2 ^ 4 ≤ 22 ^ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:n ≤ 85⊢ ↑1 * (↑85 - 1) < (↑3 + 1) * 2 ^ 3 * ↑4 * (693147 / 1000000) All goals completed! 🐙)
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:n ≤ 332⊢ ¬ShuniaCounterexample nn:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:n ≤ 332⊢ ¬ShuniaCounterexample n exact not_shuniaCounterexample_of_dyadic_small hn2 (a := 86) (b := 332) (q := 4)
(m := 6) (p := 1) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:n ≤ 332⊢ 86 ≤ n All goals completed! 🐙) h332 (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:n ≤ 332⊢ 1 ≤ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:n ≤ 332⊢ 4 ^ 4 ≤ 3 * n All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:n ≤ 332⊢ 3 * n < 4 ^ (4 + 1) All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:n ≤ 332⊢ 2 ^ 6 ≤ 86 ^ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:n ≤ 332⊢ ↑1 * (↑332 - 1) < (↑4 + 1) * 2 ^ 4 * ↑6 * (693147 / 1000000) All goals completed! 🐙)
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:n ≤ 341⊢ ¬ShuniaCounterexample nn:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:n ≤ 341⊢ ¬ShuniaCounterexample n exact not_shuniaCounterexample_of_dyadic_small hn2 (a := 333) (b := 341) (q := 4)
(m := 8) (p := 1) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:n ≤ 341⊢ 333 ≤ n All goals completed! 🐙) h341 (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:n ≤ 341⊢ 1 ≤ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:n ≤ 341⊢ 4 ^ 4 ≤ 3 * n All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:n ≤ 341⊢ 3 * n < 4 ^ (4 + 1) All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:n ≤ 341⊢ 2 ^ 8 ≤ 333 ^ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:n ≤ 341⊢ ↑1 * (↑341 - 1) < (↑4 + 1) * 2 ^ 4 * ↑8 * (693147 / 1000000) All goals completed! 🐙)
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:n ≤ 1023⊢ ¬ShuniaCounterexample nn:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:n ≤ 1023⊢ ¬ShuniaCounterexample n exact not_shuniaCounterexample_of_dyadic_small hn2 (a := 342) (b := 1023) (q := 5)
(m := 8) (p := 1) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:n ≤ 1023⊢ 342 ≤ n All goals completed! 🐙) h1023 (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:n ≤ 1023⊢ 1 ≤ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:n ≤ 1023⊢ 4 ^ 5 ≤ 3 * n All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:n ≤ 1023⊢ 3 * n < 4 ^ (5 + 1) All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:n ≤ 1023⊢ 2 ^ 8 ≤ 342 ^ 1 All goals completed! 🐙)
(n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:n ≤ 1023⊢ ↑1 * (↑1023 - 1) < (↑5 + 1) * 2 ^ 5 * ↑8 * (693147 / 1000000) All goals completed! 🐙)
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:n ≤ 1330⊢ ¬ShuniaCounterexample nn:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:¬n ≤ 1330⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:n ≤ 1330⊢ ¬ShuniaCounterexample n exact not_shuniaCounterexample_of_dyadic_small hn2 (a := 1024) (b := 1330) (q := 5)
(m := 10) (p := 1) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:n ≤ 1330⊢ 1024 ≤ n All goals completed! 🐙) h1330 (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:n ≤ 1330⊢ 1 ≤ 1 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:n ≤ 1330⊢ 4 ^ 5 ≤ 3 * n All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:n ≤ 1330⊢ 3 * n < 4 ^ (5 + 1) All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:n ≤ 1330⊢ 2 ^ 10 ≤ 1024 ^ 1 All goals completed! 🐙)
(n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:n ≤ 1330⊢ ↑1 * (↑1330 - 1) < (↑5 + 1) * 2 ^ 5 * ↑10 * (693147 / 1000000) All goals completed! 🐙)
n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:¬n ≤ 1330⊢ ¬ShuniaCounterexample n exact not_shuniaCounterexample_of_dyadic_small hn2 (a := 1331) (b := 1365) (q := 5)
(m := 31) (p := 3) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:¬n ≤ 1330⊢ 1331 ≤ n All goals completed! 🐙) hn (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:¬n ≤ 1330⊢ 1 ≤ 3 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:¬n ≤ 1330⊢ 4 ^ 5 ≤ 3 * n All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:¬n ≤ 1330⊢ 3 * n < 4 ^ (5 + 1) All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:¬n ≤ 1330⊢ 2 ^ 31 ≤ 1331 ^ 3 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 1365h3:¬n ≤ 3h5:¬n ≤ 5h17:¬n ≤ 17h21:¬n ≤ 21h85:¬n ≤ 85h332:¬n ≤ 332h341:¬n ≤ 341h1023:¬n ≤ 1023h1330:¬n ≤ 1330⊢ ↑3 * (↑1365 - 1) < (↑5 + 1) * 2 ^ 5 * ↑31 * (693147 / 1000000) All goals completed! 🐙)
General dyadic helper for 1366 ≤ n ≤ 21845 using Chebyshev.theta_ge:
if a ≤ n ≤ b, 4 ^ q ≤ 3 * n < 4 ^ (q + 1), 2 ^ m ≤ a, b ≤ S_hi ^ 2, and
(1 - 693147/1000000) * b < ((q + 1) * 2 ^ q - 2 * S_hi - 2) * m * 693147/1000000,
then ¬ ShuniaCounterexample n.
lemma not_shuniaCounterexample_of_chebyshev_dyadic {n a b q m S_hi : ℕ}
(hn2 : 2 ≤ n) (ha : a ≤ n) (hb : n ≤ b)
(hlo : 4 ^ q ≤ 3 * n) (hhi : 3 * n < 4 ^ (q + 1))
(hpow : 2 ^ m ≤ a) (hsqrt : b ≤ S_hi ^ 2)
(hW_pos : 2 * S_hi + 2 ≤ (q + 1) * 2 ^ q)
(hineq : (1 - 693147 / 1000000 : ℝ) * (b : ℝ) <
(((q + 1) * 2 ^ q : ℝ) - 2 * (S_hi : ℝ) - 2) * (m : ℝ) * (693147 / 1000000 : ℝ)) :
¬ ShuniaCounterexample n := n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)⊢ ¬ShuniaCounterexample n
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)⊢ ¬(quarterClock (n - 1) + 1) * 2 ^ quarterClock (n - 1) ≤ shuniaDeficit n
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)⊢ ¬(q + 1) * 2 ^ q ≤ shuniaDeficit n
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit n⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑n⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑n⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑nhlog_m:↑m * (693147 / 1000000) ≤ Real.log ↑n⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑nhlog_m:↑m * (693147 / 1000000) ≤ Real.log ↑nhsqrt_le:√↑n ≤ ↑S_hi⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑nhlog_m:↑m * (693147 / 1000000) ≤ Real.log ↑nhsqrt_le:√↑n ≤ ↑S_hihlog_succ:Real.log (↑n + 1) ≤ 2 * Real.log ↑n⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑nhlog_m:↑m * (693147 / 1000000) ≤ Real.log ↑nhsqrt_le:√↑n ≤ ↑S_hihlog_succ:Real.log (↑n + 1) ≤ 2 * Real.log ↑nhtheta:↑n * Real.log 2 - Real.log (↑n + 1) - 2 * √↑n * Real.log ↑n ≤ Chebyshev.theta ↑n⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑nhlog_m:↑m * (693147 / 1000000) ≤ Real.log ↑nhsqrt_le:√↑n ≤ ↑S_hihlog_succ:Real.log (↑n + 1) ≤ 2 * Real.log ↑nhtheta:↑n * Real.log 2 - Real.log (↑n + 1) - 2 * √↑n * Real.log ↑n ≤ Chebyshev.theta ↑nhlog2:0.6931471803 < Real.log 2⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑nhlog_m:↑m * (693147 / 1000000) ≤ Real.log ↑nhsqrt_le:√↑n ≤ ↑S_hihlog_succ:Real.log (↑n + 1) ≤ 2 * Real.log ↑nhtheta:↑n * Real.log 2 - Real.log (↑n + 1) - 2 * √↑n * Real.log ↑n ≤ Chebyshev.theta ↑nhlog2:0.6931471803 < Real.log 2hdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑nhlog_m:↑m * (693147 / 1000000) ≤ Real.log ↑nhsqrt_le:√↑n ≤ ↑S_hihlog_succ:Real.log (↑n + 1) ≤ 2 * Real.log ↑nhtheta:↑n * Real.log 2 - Real.log (↑n + 1) - 2 * √↑n * Real.log ↑n ≤ Chebyshev.theta ↑nhlog2:0.6931471803 < Real.log 2hdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1hce_r:↑((q + 1) * 2 ^ q) * Real.log ↑n ≤ ↑(shuniaDeficit n) * Real.log ↑n⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑nhlog_m:↑m * (693147 / 1000000) ≤ Real.log ↑nhsqrt_le:√↑n ≤ ↑S_hihlog_succ:Real.log (↑n + 1) ≤ 2 * Real.log ↑nhtheta:↑n * Real.log 2 - Real.log (↑n + 1) - 2 * √↑n * Real.log ↑n ≤ Chebyshev.theta ↑nhlog2:0.6931471803 < Real.log 2hdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1hce_r:↑((q + 1) * 2 ^ q) * Real.log ↑n ≤ ↑(shuniaDeficit n) * Real.log ↑nhn_le:↑n ≤ ↑b⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑nhlog_m:↑m * (693147 / 1000000) ≤ Real.log ↑nhsqrt_le:√↑n ≤ ↑S_hihlog_succ:Real.log (↑n + 1) ≤ 2 * Real.log ↑nhtheta:↑n * Real.log 2 - Real.log (↑n + 1) - 2 * √↑n * Real.log ↑n ≤ Chebyshev.theta ↑nhlog2:0.6931471803 < Real.log 2hdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1hce_r:↑((q + 1) * 2 ^ q) * Real.log ↑n ≤ ↑(shuniaDeficit n) * Real.log ↑nhn_le:↑n ≤ ↑bhW_r:0 ≤ (↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2⊢ False
n:ℕa:ℕb:ℕq:ℕm:ℕS_hi:ℕhn2:2 ≤ nha:a ≤ nhb:n ≤ bhlo:4 ^ q ≤ 3 * nhhi:3 * n < 4 ^ (q + 1)hpow:2 ^ m ≤ ahsqrt:b ≤ S_hi ^ 2hW_pos:2 * S_hi + 2 ≤ (q + 1) * 2 ^ qhineq:(1 - 693147 / 1000000) * ↑b < ((↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2) * ↑m * (693147 / 1000000)hce:(q + 1) * 2 ^ q ≤ shuniaDeficit nhlog_pos:0 < Real.log ↑nhpow_n:2 ^ m ≤ ↑nhlog_m:↑m * (693147 / 1000000) ≤ Real.log ↑nhsqrt_le:√↑n ≤ ↑S_hihlog_succ:Real.log (↑n + 1) ≤ 2 * Real.log ↑nhtheta:↑n * Real.log 2 - Real.log (↑n + 1) - 2 * √↑n * Real.log ↑n ≤ Chebyshev.theta ↑nhlog2:0.6931471803 < Real.log 2hdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1hn_le:↑n ≤ ↑bhW_r:0 ≤ (↑q + 1) * 2 ^ q - 2 * ↑S_hi - 2hce_r:(↑q + 1) * 2 ^ q * Real.log ↑n ≤ ↑(shuniaDeficit n) * Real.log ↑n⊢ False
All goals completed! 🐙
Unconditional Small-n Verification (2 ≤ n ≤ 21845):
No integer n in [2, 21845] satisfies ShuniaCounterexample n (0 hypotheses).
theorem not_shuniaCounterexample_of_le_21845 {n : ℕ} (hn2 : 2 ≤ n) (hn : n ≤ 21845) :
¬ ShuniaCounterexample n := n:ℕhn2:2 ≤ nhn:n ≤ 21845⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:n ≤ 1365⊢ ¬ShuniaCounterexample nn:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:n ≤ 1365⊢ ¬ShuniaCounterexample n All goals completed! 🐙
n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:n ≤ 5461⊢ ¬ShuniaCounterexample nn:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:n ≤ 5461⊢ ¬ShuniaCounterexample n exact not_shuniaCounterexample_of_chebyshev_dyadic hn2 (a := 1366) (b := 5461) (q := 6)
(m := 10) (S_hi := 74) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:n ≤ 5461⊢ 1366 ≤ n All goals completed! 🐙) h5461 (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:n ≤ 5461⊢ 4 ^ 6 ≤ 3 * n All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:n ≤ 5461⊢ 3 * n < 4 ^ (6 + 1) All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:n ≤ 5461⊢ 2 ^ 10 ≤ 1366 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:n ≤ 5461⊢ 5461 ≤ 74 ^ 2 All goals completed! 🐙)
(n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:n ≤ 5461⊢ 2 * 74 + 2 ≤ (6 + 1) * 2 ^ 6 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:n ≤ 5461⊢ (1 - 693147 / 1000000) * ↑5461 < ((↑6 + 1) * 2 ^ 6 - 2 * ↑74 - 2) * ↑10 * (693147 / 1000000) All goals completed! 🐙)
n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:n ≤ 16383⊢ ¬ShuniaCounterexample nn:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:¬n ≤ 16383⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:n ≤ 16383⊢ ¬ShuniaCounterexample n exact not_shuniaCounterexample_of_chebyshev_dyadic hn2 (a := 5462) (b := 16383) (q := 7)
(m := 12) (S_hi := 128) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:n ≤ 16383⊢ 5462 ≤ n All goals completed! 🐙) h16383 (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:n ≤ 16383⊢ 4 ^ 7 ≤ 3 * n All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:n ≤ 16383⊢ 3 * n < 4 ^ (7 + 1) All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:n ≤ 16383⊢ 2 ^ 12 ≤ 5462 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:n ≤ 16383⊢ 16383 ≤ 128 ^ 2 All goals completed! 🐙)
(n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:n ≤ 16383⊢ 2 * 128 + 2 ≤ (7 + 1) * 2 ^ 7 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:n ≤ 16383⊢ (1 - 693147 / 1000000) * ↑16383 < ((↑7 + 1) * 2 ^ 7 - 2 * ↑128 - 2) * ↑12 * (693147 / 1000000) All goals completed! 🐙)
n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:¬n ≤ 16383⊢ ¬ShuniaCounterexample n exact not_shuniaCounterexample_of_chebyshev_dyadic hn2 (a := 16384) (b := 21845) (q := 7)
(m := 14) (S_hi := 148) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:¬n ≤ 16383⊢ 16384 ≤ n All goals completed! 🐙) hn (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:¬n ≤ 16383⊢ 4 ^ 7 ≤ 3 * n All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:¬n ≤ 16383⊢ 3 * n < 4 ^ (7 + 1) All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:¬n ≤ 16383⊢ 2 ^ 14 ≤ 16384 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:¬n ≤ 16383⊢ 21845 ≤ 148 ^ 2 All goals completed! 🐙)
(n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:¬n ≤ 16383⊢ 2 * 148 + 2 ≤ (7 + 1) * 2 ^ 7 All goals completed! 🐙) (n:ℕhn2:2 ≤ nhn:n ≤ 21845h1365:¬n ≤ 1365h5461:¬n ≤ 5461h16383:¬n ≤ 16383⊢ (1 - 693147 / 1000000) * ↑21845 < ((↑7 + 1) * 2 ^ 7 - 2 * ↑148 - 2) * ↑14 * (693147 / 1000000) All goals completed! 🐙)
4. One-Sided Landau Converse (¬ RiemannHypothesis → ∃ n ≥ 2, ShuniaCounterexample n)
The one-sided Chebyshev defect g(t) = C * t ^ α + ψ(t) - t.
noncomputable def psiNegDefect (C α : ℝ) (t : ℝ) : ℝ :=
C * t ^ α + Chebyshev.psi t - ttheorem integrableOn_const_rpow_add_psi_sub_self_mul_cpow {s : ℂ} {C α : ℝ}
(hα : α < s.re) (hs : 1 < s.re) :
IntegrableOn (fun t : ℝ ↦
((C * t ^ α + Chebyshev.psi t - t : ℝ) : ℂ) * (t : ℂ) ^ (-(s + 1))) (Set.Ioi 1) := s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.re⊢ IntegrableOn (fun t ↦ ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))) (Set.Ioi 1) volume
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehint_rpow:IntegrableOn (fun t ↦ ↑(C * t ^ α) * ↑t ^ (-(s + 1))) (Set.Ioi 1) volume⊢ IntegrableOn (fun t ↦ ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))) (Set.Ioi 1) volume
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehint_rpow:IntegrableOn (fun t ↦ ↑(C * t ^ α) * ↑t ^ (-(s + 1))) (Set.Ioi 1) volumet:ℝx✝:t ∈ Set.Ioi 1⊢ (((fun t ↦ ↑(C * t ^ α) * ↑t ^ (-(s + 1))) + fun t ↦ ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))) - fun t ↦
↑t * ↑t ^ (-(s + 1)))
t =
↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehint_rpow:IntegrableOn (fun t ↦ ↑(C * t ^ α) * ↑t ^ (-(s + 1))) (Set.Ioi 1) volumet:ℝx✝:t ∈ Set.Ioi 1⊢ ↑(C * t ^ α) * ↑t ^ (-(s + 1)) + ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1)) - ↑t * ↑t ^ (-(s + 1)) =
↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehint_rpow:IntegrableOn (fun t ↦ ↑(C * t ^ α) * ↑t ^ (-(s + 1))) (Set.Ioi 1) volumet:ℝx✝:t ∈ Set.Ioi 1⊢ ↑C * ↑(t ^ α) * ↑t ^ (-(s + 1)) + ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1)) - ↑t * ↑t ^ (-(s + 1)) =
(↑C * ↑(t ^ α) + ↑(Chebyshev.psi t) - ↑t) * ↑t ^ (-(s + 1))
All goals completed! 🐙theorem mellin_const_rpow_add_psi_sub_self {s : ℂ} {C α : ℝ} (hα : α < s.re) (hs : 1 < s.re) :
(C : ℂ) / (s - α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
∫ t in Set.Ioi (1 : ℝ),
((C * t ^ α + Chebyshev.psi t - t : ℝ) : ℂ) * (t : ℂ) ^ (-(s + 1)) := s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.re⊢ ↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehs0:s ≠ 0⊢ ↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehs0:s ≠ 0hpsi:-deriv riemannZeta s / riemannZeta s = s * ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))⊢ ↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehs0:s ≠ 0hpsi:-deriv riemannZeta s / riemannZeta s = s * ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hpsi_div:-deriv riemannZeta s / (s * riemannZeta s) = ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))⊢ ↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehs0:s ≠ 0hpsi:-deriv riemannZeta s / riemannZeta s = s * ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hpsi_div:-deriv riemannZeta s / (s * riemannZeta s) = ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hrpow:↑C / (s - ↑α) = ∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α) * ↑t ^ (-(s + 1))⊢ ↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehs0:s ≠ 0hpsi:-deriv riemannZeta s / riemannZeta s = s * ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hpsi_div:-deriv riemannZeta s / (s * riemannZeta s) = ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hrpow:↑C / (s - ↑α) = ∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α) * ↑t ^ (-(s + 1))hself:1 / (s - 1) = ∫ (t : ℝ) in Set.Ioi 1, ↑t * ↑t ^ (-(s + 1))⊢ ↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehs0:s ≠ 0hpsi:-deriv riemannZeta s / riemannZeta s = s * ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hpsi_div:-deriv riemannZeta s / (s * riemannZeta s) = ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hrpow:↑C / (s - ↑α) = ∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α) * ↑t ^ (-(s + 1))hself:1 / (s - 1) = ∫ (t : ℝ) in Set.Ioi 1, ↑t * ↑t ^ (-(s + 1))hint_rpow:IntegrableOn (fun t ↦ ↑(C * t ^ α) * ↑t ^ (-(s + 1))) (Set.Ioi 1) volume⊢ ↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehs0:s ≠ 0hpsi:-deriv riemannZeta s / riemannZeta s = s * ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hpsi_div:-deriv riemannZeta s / (s * riemannZeta s) = ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hrpow:↑C / (s - ↑α) = ∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α) * ↑t ^ (-(s + 1))hself:1 / (s - 1) = ∫ (t : ℝ) in Set.Ioi 1, ↑t * ↑t ^ (-(s + 1))hint_rpow:IntegrableOn (fun t ↦ ↑(C * t ^ α) * ↑t ^ (-(s + 1))) (Set.Ioi 1) volumehsub:↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
↑C / (s - ↑α) + -deriv riemannZeta s / (s * riemannZeta s) - 1 / (s - 1)⊢ ↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehs0:s ≠ 0hpsi:-deriv riemannZeta s / riemannZeta s = s * ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hpsi_div:-deriv riemannZeta s / (s * riemannZeta s) = ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hrpow:↑C / (s - ↑α) = ∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α) * ↑t ^ (-(s + 1))hself:1 / (s - 1) = ∫ (t : ℝ) in Set.Ioi 1, ↑t * ↑t ^ (-(s + 1))hint_rpow:IntegrableOn (fun t ↦ ↑(C * t ^ α) * ↑t ^ (-(s + 1))) (Set.Ioi 1) volumehsub:↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
↑C / (s - ↑α) + -deriv riemannZeta s / (s * riemannZeta s) - 1 / (s - 1)⊢ (∫ (a : ℝ) in Set.Ioi 1, ↑(C * a ^ α) * ↑a ^ (-(s + 1)) + ↑(Chebyshev.psi a) * ↑a ^ (-(s + 1))) -
∫ (t : ℝ) in Set.Ioi 1, ↑t * ↑t ^ (-(s + 1)) =
∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehs0:s ≠ 0hpsi:-deriv riemannZeta s / riemannZeta s = s * ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hpsi_div:-deriv riemannZeta s / (s * riemannZeta s) = ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hrpow:↑C / (s - ↑α) = ∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α) * ↑t ^ (-(s + 1))hself:1 / (s - 1) = ∫ (t : ℝ) in Set.Ioi 1, ↑t * ↑t ^ (-(s + 1))hint_rpow:IntegrableOn (fun t ↦ ↑(C * t ^ α) * ↑t ^ (-(s + 1))) (Set.Ioi 1) volumehsub:↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
↑C / (s - ↑α) + -deriv riemannZeta s / (s * riemannZeta s) - 1 / (s - 1)hint_add:IntegrableOn (fun a ↦ ↑(C * a ^ α) * ↑a ^ (-(s + 1)) + ↑(Chebyshev.psi a) * ↑a ^ (-(s + 1))) (Set.Ioi 1) volume⊢ (∫ (a : ℝ) in Set.Ioi 1, ↑(C * a ^ α) * ↑a ^ (-(s + 1)) + ↑(Chebyshev.psi a) * ↑a ^ (-(s + 1))) -
∫ (t : ℝ) in Set.Ioi 1, ↑t * ↑t ^ (-(s + 1)) =
∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehs0:s ≠ 0hpsi:-deriv riemannZeta s / riemannZeta s = s * ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hpsi_div:-deriv riemannZeta s / (s * riemannZeta s) = ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hrpow:↑C / (s - ↑α) = ∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α) * ↑t ^ (-(s + 1))hself:1 / (s - 1) = ∫ (t : ℝ) in Set.Ioi 1, ↑t * ↑t ^ (-(s + 1))hint_rpow:IntegrableOn (fun t ↦ ↑(C * t ^ α) * ↑t ^ (-(s + 1))) (Set.Ioi 1) volumehsub:↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
↑C / (s - ↑α) + -deriv riemannZeta s / (s * riemannZeta s) - 1 / (s - 1)hint_add:IntegrableOn (fun a ↦ ↑(C * a ^ α) * ↑a ^ (-(s + 1)) + ↑(Chebyshev.psi a) * ↑a ^ (-(s + 1))) (Set.Ioi 1) volume⊢ ∫ (a : ℝ) in Set.Ioi 1, ↑(C * a ^ α) * ↑a ^ (-(s + 1)) + ↑(Chebyshev.psi a) * ↑a ^ (-(s + 1)) - ↑a * ↑a ^ (-(s + 1)) =
∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehs0:s ≠ 0hpsi:-deriv riemannZeta s / riemannZeta s = s * ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hpsi_div:-deriv riemannZeta s / (s * riemannZeta s) = ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hrpow:↑C / (s - ↑α) = ∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α) * ↑t ^ (-(s + 1))hself:1 / (s - 1) = ∫ (t : ℝ) in Set.Ioi 1, ↑t * ↑t ^ (-(s + 1))hint_rpow:IntegrableOn (fun t ↦ ↑(C * t ^ α) * ↑t ^ (-(s + 1))) (Set.Ioi 1) volumehsub:↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
↑C / (s - ↑α) + -deriv riemannZeta s / (s * riemannZeta s) - 1 / (s - 1)hint_add:IntegrableOn (fun a ↦ ↑(C * a ^ α) * ↑a ^ (-(s + 1)) + ↑(Chebyshev.psi a) * ↑a ^ (-(s + 1))) (Set.Ioi 1) volumet:ℝx✝:t ∈ Set.Ioi 1⊢ ↑(C * t ^ α) * ↑t ^ (-(s + 1)) + ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1)) - ↑t * ↑t ^ (-(s + 1)) =
↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
s:ℂC:ℝα:ℝhα:α < s.rehs:1 < s.rehs0:s ≠ 0hpsi:-deriv riemannZeta s / riemannZeta s = s * ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hpsi_div:-deriv riemannZeta s / (s * riemannZeta s) = ∫ (t : ℝ) in Set.Ioi 1, ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1))hrpow:↑C / (s - ↑α) = ∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α) * ↑t ^ (-(s + 1))hself:1 / (s - 1) = ∫ (t : ℝ) in Set.Ioi 1, ↑t * ↑t ^ (-(s + 1))hint_rpow:IntegrableOn (fun t ↦ ↑(C * t ^ α) * ↑t ^ (-(s + 1))) (Set.Ioi 1) volumehsub:↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
↑C / (s - ↑α) + -deriv riemannZeta s / (s * riemannZeta s) - 1 / (s - 1)hint_add:IntegrableOn (fun a ↦ ↑(C * a ^ α) * ↑a ^ (-(s + 1)) + ↑(Chebyshev.psi a) * ↑a ^ (-(s + 1))) (Set.Ioi 1) volumet:ℝx✝:t ∈ Set.Ioi 1⊢ ↑C * ↑(t ^ α) * ↑t ^ (-(s + 1)) + ↑(Chebyshev.psi t) * ↑t ^ (-(s + 1)) - ↑t * ↑t ^ (-(s + 1)) =
(↑C * ↑(t ^ α) + ↑(Chebyshev.psi t) - ↑t) * ↑t ^ (-(s + 1))
All goals completed! 🐙theorem mellin_neg_lhs_eq_completedPoleZeta {s : ℂ} {C α : ℝ}
(hs0 : s ≠ 0) (hs1 : s ≠ 1) (hz : riemannZeta s ≠ 0) :
(C : ℂ) / (s - α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
(C : ℂ) / (s - α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s) := s:ℂC:ℝα:ℝhs0:s ≠ 0hs1:s ≠ 1hz:riemannZeta s ≠ 0⊢ ↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
↑C / (s - ↑α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s)
s:ℂC:ℝα:ℝhs0:s ≠ 0hs1:s ≠ 1hz:riemannZeta s ≠ 0h1:deriv completedPoleZeta s / (s * completedPoleZeta s) = deriv completedPoleZeta s / completedPoleZeta s / s⊢ ↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
↑C / (s - ↑α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s)
s:ℂC:ℝα:ℝhs0:s ≠ 0hs1:s ≠ 1hz:riemannZeta s ≠ 0h1:deriv completedPoleZeta s / (s * completedPoleZeta s) = deriv completedPoleZeta s / completedPoleZeta s / s⊢ ↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
↑C / (s - ↑α) - 1 / s - (1 / (s - 1) + deriv riemannZeta s / riemannZeta s) / s
s:ℂC:ℝα:ℝhs0:s ≠ 0hs1:s ≠ 1hz:riemannZeta s ≠ 0h1:deriv completedPoleZeta s / (s * completedPoleZeta s) = deriv completedPoleZeta s / completedPoleZeta s / shs1':s - 1 ≠ 0⊢ ↑C / (s - ↑α) - 1 / (s - 1) - deriv riemannZeta s / (s * riemannZeta s) =
↑C / (s - ↑α) - 1 / s - (1 / (s - 1) + deriv riemannZeta s / riemannZeta s) / s
s:ℂC:ℝα:ℝhs0:s ≠ 0hs1:s ≠ 1hz:riemannZeta s ≠ 0h1:deriv completedPoleZeta s / (s * completedPoleZeta s) = deriv completedPoleZeta s / completedPoleZeta s / shs1':s - 1 ≠ 0⊢ s * (↑C * (s - 1) / (s - ↑α) - 1) * riemannZeta s - (s - 1) * deriv riemannZeta s =
(s - 1) * riemannZeta s * (↑C * s / (s - ↑α) - 1) - (riemannZeta s + (s - 1) * deriv riemannZeta s)
All goals completed! 🐙theorem integrableOn_psiNegDefect_rpow_of_one_lt {C α σ : ℝ} (hα : α < σ) (hσ : 1 < σ) :
IntegrableOn (fun t : ℝ ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) := C:ℝα:ℝσ:ℝhα:α < σhσ:1 < σ⊢ IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume
C:ℝα:ℝσ:ℝhα:α < σhσ:1 < σhα_c:α < (↑σ).re⊢ IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume
C:ℝα:ℝσ:ℝhα:α < σhσ:1 < σhα_c:α < (↑σ).rehσ_c:1 < (↑σ).re⊢ IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume
C:ℝα:ℝσ:ℝhα:α < σhσ:1 < σhα_c:α < (↑σ).rehσ_c:1 < (↑σ).reh_cpow:IntegrableOn (fun t ↦ ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))) (Set.Ioi 1) volume⊢ IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume
C:ℝα:ℝσ:ℝhα:α < σhσ:1 < σhα_c:α < (↑σ).rehσ_c:1 < (↑σ).reh_cpow:IntegrableOn (fun t ↦ ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))) (Set.Ioi 1) volumeh_re:IntegrableOn (fun t ↦ (↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))).re) (Set.Ioi 1) volume⊢ IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume
C:ℝα:ℝσ:ℝhα:α < σhσ:1 < σhα_c:α < (↑σ).rehσ_c:1 < (↑σ).reh_cpow:IntegrableOn (fun t ↦ ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))) (Set.Ioi 1) volumeh_re:IntegrableOn (fun t ↦ (↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))).re) (Set.Ioi 1) volumet:ℝht:t ∈ Set.Ioi 1⊢ (fun t ↦ (↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))).re) t = psiNegDefect C α t * t ^ (-(σ + 1))
C:ℝα:ℝσ:ℝhα:α < σhσ:1 < σhα_c:α < (↑σ).rehσ_c:1 < (↑σ).reh_cpow:IntegrableOn (fun t ↦ ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))) (Set.Ioi 1) volumeh_re:IntegrableOn (fun t ↦ (↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))).re) (Set.Ioi 1) volumet:ℝht:t ∈ Set.Ioi 1ht_pos:0 < t⊢ (fun t ↦ (↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))).re) t = psiNegDefect C α t * t ^ (-(σ + 1))
C:ℝα:ℝσ:ℝhα:α < σhσ:1 < σhα_c:α < (↑σ).rehσ_c:1 < (↑σ).reh_cpow:IntegrableOn (fun t ↦ ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))) (Set.Ioi 1) volumeh_re:IntegrableOn (fun t ↦ (↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))).re) (Set.Ioi 1) volumet:ℝht:t ∈ Set.Ioi 1ht_pos:0 < th_exp:-(↑σ + 1) = ↑(-(σ + 1))⊢ (fun t ↦ (↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))).re) t = psiNegDefect C α t * t ^ (-(σ + 1))
C:ℝα:ℝσ:ℝhα:α < σhσ:1 < σhα_c:α < (↑σ).rehσ_c:1 < (↑σ).reh_cpow:IntegrableOn (fun t ↦ ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))) (Set.Ioi 1) volumeh_re:IntegrableOn (fun t ↦ (↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))).re) (Set.Ioi 1) volumet:ℝht:t ∈ Set.Ioi 1ht_pos:0 < th_exp:-(↑σ + 1) = ↑(-(σ + 1))⊢ (↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(↑σ + 1))).re = (C * t ^ α + Chebyshev.psi t - t) * t ^ (-(σ + 1))
All goals completed! 🐙theorem mellinLogDeriv_zero_psiNegDefect_eq_of_one_lt_re {C α : ℝ} {s : ℂ}
(hα : α < s.re) (hs : 1 < s.re) :
mellinLogDeriv (psiNegDefect C α) 0 s =
(C : ℂ) / (s - α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s) := C:ℝα:ℝs:ℂhα:α < s.rehs:1 < s.re⊢ mellinLogDeriv (psiNegDefect C α) 0 s = ↑C / (s - ↑α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s)
C:ℝα:ℝs:ℂhα:α < s.rehs:1 < s.rehs0:s ≠ 0⊢ mellinLogDeriv (psiNegDefect C α) 0 s = ↑C / (s - ↑α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s)
C:ℝα:ℝs:ℂhα:α < s.rehs:1 < s.rehs0:s ≠ 0hs1:s ≠ 1⊢ mellinLogDeriv (psiNegDefect C α) 0 s = ↑C / (s - ↑α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s)
C:ℝα:ℝs:ℂhα:α < s.rehs:1 < s.rehs0:s ≠ 0hs1:s ≠ 1hz:riemannZeta s ≠ 0⊢ mellinLogDeriv (psiNegDefect C α) 0 s = ↑C / (s - ↑α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s)
C:ℝα:ℝs:ℂhα:α < s.rehs:1 < s.rehs0:s ≠ 0hs1:s ≠ 1hz:riemannZeta s ≠ 0⊢ ∫ (t : ℝ) in Set.Ioi 1, (-1) ^ 0 * ↑(psiNegDefect C α t * Real.log t ^ 0) * ↑t ^ (-(s + 1)) =
∫ (t : ℝ) in Set.Ioi 1, ↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
C:ℝα:ℝs:ℂhα:α < s.rehs:1 < s.rehs0:s ≠ 0hs1:s ≠ 1hz:riemannZeta s ≠ 0t:ℝx✝:t ∈ Set.Ioi 1⊢ (-1) ^ 0 * ↑(psiNegDefect C α t * Real.log t ^ 0) * ↑t ^ (-(s + 1)) =
↑(C * t ^ α + Chebyshev.psi t - t) * ↑t ^ (-(s + 1))
All goals completed! 🐙theorem completedPoleZeta_mul_mellinLogDeriv_zero_psiNegDefect_eq {C α σ₀ : ℝ}
(hα0 : 0 < α) (hασ₀ : α ≤ σ₀) (hσ₀1 : σ₀ ≤ 1)
(hg_nonneg : ∀ t : ℝ, 1 ≤ t → 0 ≤ psiNegDefect C α t)
(hInt : ∀ σ : ℝ, σ₀ < σ →
IntegrableOn (fun t : ℝ ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1))
{s : ℂ} (hs : σ₀ < s.re) :
completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
((C : ℂ) / (s - α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s := C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.re⊢ completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s
C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.reU:Set ℂ := {z | σ₀ < z.re}⊢ completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s
C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.reU:Set ℂ := {z | σ₀ < z.re}hOpen:IsOpen U⊢ completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s
C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.reU:Set ℂ := {z | σ₀ < z.re}hOpen:IsOpen UhPre:IsPreconnected U⊢ completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s
C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.reU:Set ℂ := {z | σ₀ < z.re}hOpen:IsOpen UhPre:IsPreconnected UG_lhs:ℂ → ℂ := fun z ↦ completedPoleZeta z * mellinLogDeriv (psiNegDefect C α) 0 z⊢ completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s
C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.reU:Set ℂ := {z | σ₀ < z.re}hOpen:IsOpen UhPre:IsPreconnected UG_lhs:ℂ → ℂ := fun z ↦ completedPoleZeta z * mellinLogDeriv (psiNegDefect C α) 0 zG_rhs:ℂ → ℂ := fun z ↦ (↑C / (z - ↑α) - 1 / z) * completedPoleZeta z - deriv completedPoleZeta z / z⊢ completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s
C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.reU:Set ℂ := {z | σ₀ < z.re}hOpen:IsOpen UhPre:IsPreconnected UG_lhs:ℂ → ℂ := fun z ↦ completedPoleZeta z * mellinLogDeriv (psiNegDefect C α) 0 zG_rhs:ℂ → ℂ := fun z ↦ (↑C / (z - ↑α) - 1 / z) * completedPoleZeta z - deriv completedPoleZeta z / zhAn_lhs:AnalyticOnNhd ℂ G_lhs U⊢ completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s
C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.reU:Set ℂ := {z | σ₀ < z.re}hOpen:IsOpen UhPre:IsPreconnected UG_lhs:ℂ → ℂ := fun z ↦ completedPoleZeta z * mellinLogDeriv (psiNegDefect C α) 0 zG_rhs:ℂ → ℂ := fun z ↦ (↑C / (z - ↑α) - 1 / z) * completedPoleZeta z - deriv completedPoleZeta z / zhAn_lhs:AnalyticOnNhd ℂ G_lhs UhAn_rhs:AnalyticOnNhd ℂ G_rhs U⊢ completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s
C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.reU:Set ℂ := {z | σ₀ < z.re}hOpen:IsOpen UhPre:IsPreconnected UG_lhs:ℂ → ℂ := fun z ↦ completedPoleZeta z * mellinLogDeriv (psiNegDefect C α) 0 zG_rhs:ℂ → ℂ := fun z ↦ (↑C / (z - ↑α) - 1 / z) * completedPoleZeta z - deriv completedPoleZeta z / zhAn_lhs:AnalyticOnNhd ℂ G_lhs UhAn_rhs:AnalyticOnNhd ℂ G_rhs Uh2_mem:2 ∈ U⊢ completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s
C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.reU:Set ℂ := {z | σ₀ < z.re}hOpen:IsOpen UhPre:IsPreconnected UG_lhs:ℂ → ℂ := fun z ↦ completedPoleZeta z * mellinLogDeriv (psiNegDefect C α) 0 zG_rhs:ℂ → ℂ := fun z ↦ (↑C / (z - ↑α) - 1 / z) * completedPoleZeta z - deriv completedPoleZeta z / zhAn_lhs:AnalyticOnNhd ℂ G_lhs UhAn_rhs:AnalyticOnNhd ℂ G_rhs Uh2_mem:2 ∈ UhEv:G_lhs =ᶠ[𝓝 2] G_rhs⊢ completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s
All goals completed! 🐙theorem mellinLogDeriv_zero_psiNegDefect_eq_of_completedPoleZeta_ne_zero {C α σ₀ : ℝ}
(hα0 : 0 < α) (hασ₀ : α ≤ σ₀) (hσ₀1 : σ₀ ≤ 1)
(hg_nonneg : ∀ t : ℝ, 1 ≤ t → 0 ≤ psiNegDefect C α t)
(hInt : ∀ σ : ℝ, σ₀ < σ →
IntegrableOn (fun t : ℝ ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1))
{s : ℂ} (hs : σ₀ < s.re) (hG : completedPoleZeta s ≠ 0) :
mellinLogDeriv (psiNegDefect C α) 0 s =
(C : ℂ) / (s - α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s) := C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.rehG:completedPoleZeta s ≠ 0⊢ mellinLogDeriv (psiNegDefect C α) 0 s = ↑C / (s - ↑α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s)
C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.rehG:completedPoleZeta s ≠ 0h_cleared:completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s⊢ mellinLogDeriv (psiNegDefect C α) 0 s = ↑C / (s - ↑α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s)
C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.rehG:completedPoleZeta s ≠ 0h_cleared:completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / shs0:s ≠ 0⊢ mellinLogDeriv (psiNegDefect C α) 0 s = ↑C / (s - ↑α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s)
calc
mellinLogDeriv (psiNegDefect C α) 0 s
= (completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s) / completedPoleZeta s := C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.rehG:completedPoleZeta s ≠ 0h_cleared:completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / shs0:s ≠ 0⊢ mellinLogDeriv (psiNegDefect C α) 0 s =
completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s / completedPoleZeta s
All goals completed! 🐙
_ = (((C : ℂ) / (s - α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s) /
completedPoleZeta s := C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.rehG:completedPoleZeta s ≠ 0h_cleared:completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / shs0:s ≠ 0⊢ completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s / completedPoleZeta s =
((↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s) / completedPoleZeta s
All goals completed! 🐙
_ = (C : ℂ) / (s - α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s) := C:ℝα:ℝσ₀:ℝhα0:0 < αhασ₀:α ≤ σ₀hσ₀1:σ₀ ≤ 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thInt:∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumes:ℂhs:σ₀ < s.rehG:completedPoleZeta s ≠ 0h_cleared:completedPoleZeta s * mellinLogDeriv (psiNegDefect C α) 0 s =
(↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / shs0:s ≠ 0⊢ ((↑C / (s - ↑α) - 1 / s) * completedPoleZeta s - deriv completedPoleZeta s / s) / completedPoleZeta s =
↑C / (s - ↑α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s)
All goals completed! 🐙
Pole-free analytic RHS (C : ℂ) / (s - α) - 1 / s - deriv completedPoleZeta s / (s * G s).
noncomputable def poleFreeNegRHS (C α : ℝ) (s : ℂ) : ℂ :=
(C : ℂ) / (s - α) - 1 / s - deriv completedPoleZeta s / (s * completedPoleZeta s)
Domain of holomorphy {s : ℂ | s - α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0} is open and
poleFreeNegRHS C α is differentiable on it.
lemma isOpen_and_differentiableOn_poleFreeNegRHS (C α : ℝ) :
let W : Set ℂ := {s : ℂ | s - (α : ℂ) ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}
IsOpen W ∧ DifferentiableOn ℂ (poleFreeNegRHS C α) W := C:ℝα:ℝ⊢ let W := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0};
IsOpen W ∧ DifferentiableOn ℂ (poleFreeNegRHS C α) W
C:ℝα:ℝW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}⊢ IsOpen W ∧ DifferentiableOn ℂ (poleFreeNegRHS C α) W
C:ℝα:ℝW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}⊢ IsOpen WC:ℝα:ℝW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}⊢ DifferentiableOn ℂ (poleFreeNegRHS C α) W
C:ℝα:ℝW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}⊢ IsOpen W C:ℝα:ℝW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}⊢ IsOpen fun s ↦ s ≠ 0 ∧ completedPoleZeta s ≠ 0
All goals completed! 🐙
C:ℝα:ℝW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}⊢ DifferentiableOn ℂ (poleFreeNegRHS C α) W C:ℝα:ℝW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}s:ℂhs:s ∈ W⊢ DifferentiableWithinAt ℂ (poleFreeNegRHS C α) W s
C:ℝα:ℝW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}s:ℂhs_α:s - ↑α ≠ 0hs_0:s ≠ 0hs_G:completedPoleZeta s ≠ 0⊢ DifferentiableWithinAt ℂ (poleFreeNegRHS C α) W s
C:ℝα:ℝW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}s:ℂhs_α:s - ↑α ≠ 0hs_0:s ≠ 0hs_G:completedPoleZeta s ≠ 0hd1:DifferentiableAt ℂ (fun w ↦ ↑C / (w - ↑α) - 1 / w) s⊢ DifferentiableWithinAt ℂ (poleFreeNegRHS C α) W s
C:ℝα:ℝW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}s:ℂhs_α:s - ↑α ≠ 0hs_0:s ≠ 0hs_G:completedPoleZeta s ≠ 0hd1:DifferentiableAt ℂ (fun w ↦ ↑C / (w - ↑α) - 1 / w) shd2:DifferentiableAt ℂ (fun w ↦ deriv completedPoleZeta w / (w * completedPoleZeta w)) s⊢ DifferentiableWithinAt ℂ (poleFreeNegRHS C α) W s
All goals completed! 🐙
Landau Continuation for psiNegDefect:
If 0 ≤ C * t ^ α + ψ(t) - t for all t ≥ 1, then its Mellin integral converges down to α.
theorem integrableOn_psiNegDefect_rpow_of_gt_alpha {C α : ℝ}
(hα0 : 0 < α) (hα1 : α < 1)
(hg_nonneg : ∀ t : ℝ, 1 ≤ t → 0 ≤ psiNegDefect C α t) :
∀ σ : ℝ, α < σ →
IntegrableOn (fun t : ℝ ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) := C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α t⊢ ∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}⊢ ∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ S⊢ ∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.Nonempty⊢ ∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow S⊢ ∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf S⊢ ∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_star⊢ ∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1⊢ ∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ S⊢ ∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ S⊢ σ_star = α
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = α⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_star⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_star⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_starW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_starW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hW_open:IsOpen {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hF_diff_W:DifferentiableOn ℂ (poleFreeNegRHS C α) {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_starW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hW_open:IsOpen {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hF_diff_W:DifferentiableOn ℂ (poleFreeNegRHS C α) {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hz₀_mem:↑σ_star ∈ W⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_starW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hW_open:IsOpen {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hF_diff_W:DifferentiableOn ℂ (poleFreeNegRHS C α) {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hz₀_mem:↑σ_star ∈ Wr₀:ℝhr₀_pos:r₀ > 0hr₀_sub:Metric.ball (↑σ_star) r₀ ⊆ {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_starW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hW_open:IsOpen {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hF_diff_W:DifferentiableOn ℂ (poleFreeNegRHS C α) {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hz₀_mem:↑σ_star ∈ Wr₀:ℝhr₀_pos:r₀ > 0hr₀_sub:Metric.ball (↑σ_star) r₀ ⊆ {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}r:ℝ := min (r₀ / 3) ((σ_star - α) / 2)⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_starW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hW_open:IsOpen {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hF_diff_W:DifferentiableOn ℂ (poleFreeNegRHS C α) {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hz₀_mem:↑σ_star ∈ Wr₀:ℝhr₀_pos:r₀ > 0hr₀_sub:Metric.ball (↑σ_star) r₀ ⊆ {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}r:ℝ := min (r₀ / 3) ((σ_star - α) / 2)hr_pos:0 < r⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_starW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hW_open:IsOpen {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hF_diff_W:DifferentiableOn ℂ (poleFreeNegRHS C α) {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hz₀_mem:↑σ_star ∈ Wr₀:ℝhr₀_pos:r₀ > 0hr₀_sub:Metric.ball (↑σ_star) r₀ ⊆ {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}r:ℝ := min (r₀ / 3) ((σ_star - α) / 2)hr_pos:0 < rhr_lt_r₀:r ≤ r₀ / 3⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_starW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hW_open:IsOpen {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hF_diff_W:DifferentiableOn ℂ (poleFreeNegRHS C α) {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hz₀_mem:↑σ_star ∈ Wr₀:ℝhr₀_pos:r₀ > 0hr₀_sub:Metric.ball (↑σ_star) r₀ ⊆ {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}r:ℝ := min (r₀ / 3) ((σ_star - α) / 2)hr_pos:0 < rhr_lt_r₀:r ≤ r₀ / 3hr_le_sub:r ≤ (σ_star - α) / 2⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_starW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hW_open:IsOpen {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hF_diff_W:DifferentiableOn ℂ (poleFreeNegRHS C α) {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hz₀_mem:↑σ_star ∈ Wr₀:ℝhr₀_pos:r₀ > 0hr₀_sub:Metric.ball (↑σ_star) r₀ ⊆ {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}r:ℝ := min (r₀ / 3) ((σ_star - α) / 2)hr_pos:0 < rhr_lt_r₀:r ≤ r₀ / 3hr_le_sub:r ≤ (σ_star - α) / 2c:ℝ := σ_star + r / 2⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_starW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hW_open:IsOpen {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hF_diff_W:DifferentiableOn ℂ (poleFreeNegRHS C α) {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hz₀_mem:↑σ_star ∈ Wr₀:ℝhr₀_pos:r₀ > 0hr₀_sub:Metric.ball (↑σ_star) r₀ ⊆ {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}r:ℝ := min (r₀ / 3) ((σ_star - α) / 2)hr_pos:0 < rhr_lt_r₀:r ≤ r₀ / 3hr_le_sub:r ≤ (σ_star - α) / 2c:ℝ := σ_star + r / 2hc_gt:σ_star < c⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_starW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hW_open:IsOpen {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hF_diff_W:DifferentiableOn ℂ (poleFreeNegRHS C α) {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hz₀_mem:↑σ_star ∈ Wr₀:ℝhr₀_pos:r₀ > 0hr₀_sub:Metric.ball (↑σ_star) r₀ ⊆ {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}r:ℝ := min (r₀ / 3) ((σ_star - α) / 2)hr_pos:0 < rhr_lt_r₀:r ≤ r₀ / 3hr_le_sub:r ≤ (σ_star - α) / 2c:ℝ := σ_star + r / 2hc_gt:σ_star < chball_sub:Metric.ball (↑c) r ⊆ W⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_starW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hW_open:IsOpen {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hF_diff_W:DifferentiableOn ℂ (poleFreeNegRHS C α) {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hz₀_mem:↑σ_star ∈ Wr₀:ℝhr₀_pos:r₀ > 0hr₀_sub:Metric.ball (↑σ_star) r₀ ⊆ {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}r:ℝ := min (r₀ / 3) ((σ_star - α) / 2)hr_pos:0 < rhr_lt_r₀:r ≤ r₀ / 3hr_le_sub:r ≤ (σ_star - α) / 2c:ℝ := σ_star + r / 2hc_gt:σ_star < chball_sub:Metric.ball (↑c) r ⊆ Whc_mem_W:↑c ∈ W⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_starW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hW_open:IsOpen {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hF_diff_W:DifferentiableOn ℂ (poleFreeNegRHS C α) {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hz₀_mem:↑σ_star ∈ Wr₀:ℝhr₀_pos:r₀ > 0hr₀_sub:Metric.ball (↑σ_star) r₀ ⊆ {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}r:ℝ := min (r₀ / 3) ((σ_star - α) / 2)hr_pos:0 < rhr_lt_r₀:r ≤ r₀ / 3hr_le_sub:r ≤ (σ_star - α) / 2c:ℝ := σ_star + r / 2hc_gt:σ_star < chball_sub:Metric.ball (↑c) r ⊆ Whc_mem_W:↑c ∈ WhEq:poleFreeNegRHS C α =ᶠ[𝓝 ↑c] mellinLogDeriv (psiNegDefect C α) 0⊢ False
C:ℝα:ℝhα0:0 < αhα1:α < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α tS:Set ℝ := {σ₀ | α ≤ σ₀ ∧ ∀ (σ : ℝ), σ₀ < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume}h1_mem:1 ∈ ShS_nonempty:S.NonemptyhS_bdd:BddBelow Sσ_star:ℝ := sInf Shσ_star_ge:α ≤ σ_starhσ_star_le:σ_star ≤ 1hσ_star_mem:σ_star ∈ Sh_ne:¬σ_star = αh_gt:α < σ_starhσ_star_pos:0 < σ_starW:Set ℂ := {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hW_open:IsOpen {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hF_diff_W:DifferentiableOn ℂ (poleFreeNegRHS C α) {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}hz₀_mem:↑σ_star ∈ Wr₀:ℝhr₀_pos:r₀ > 0hr₀_sub:Metric.ball (↑σ_star) r₀ ⊆ {s | s - ↑α ≠ 0 ∧ s ≠ 0 ∧ completedPoleZeta s ≠ 0}r:ℝ := min (r₀ / 3) ((σ_star - α) / 2)hr_pos:0 < rhr_lt_r₀:r ≤ r₀ / 3hr_le_sub:r ≤ (σ_star - α) / 2c:ℝ := σ_star + r / 2hc_gt:σ_star < chball_sub:Metric.ball (↑c) r ⊆ Whc_mem_W:↑c ∈ WhEq:poleFreeNegRHS C α =ᶠ[𝓝 ↑c] mellinLogDeriv (psiNegDefect C α) 0h_ext_mem:σ_star - r / 2 ∈ S⊢ False
All goals completed! 🐙lemma self_sub_psi_le_const_mul_rpow_of_forall_nat {K : ℝ} (hK : 0 < K)
(h : ∀ x : ℕ, 253 < x → (x : ℝ) - Chebyshev.psi x ≤ K * Real.sqrt x * (Real.log x) ^ 2)
{α : ℝ} (hα : 1 / 2 < α) :
∃ C : ℝ, 0 < C ∧ ∀ t : ℝ, 1 ≤ t → t - Chebyshev.psi t ≤ C * t ^ α := K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < α⊢ ∃ C, 0 < C ∧ ∀ (t : ℝ), 1 ≤ t → t - Chebyshev.psi t ≤ C * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2⊢ ∃ C, 0 < C ∧ ∀ (t : ℝ), 1 ≤ t → t - Chebyshev.psi t ≤ C * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < ε⊢ ∃ C, 0 < C ∧ ∀ (t : ℝ), 1 ≤ t → t - Chebyshev.psi t ≤ C * t ^ α
refine ⟨K * (1 / ε) ^ 2 + 255, K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < ε⊢ 0 < K * (1 / ε) ^ 2 + 255 All goals completed! 🐙, fun t ht ↦ ?_⟩
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ t⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < t⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / ε⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log t⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ α⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:253 < ⌊t⌋₊⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ αK:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:¬253 < ⌊t⌋₊⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:253 < ⌊t⌋₊⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:253 < ⌊t⌋₊h_nat:↑⌊t⌋₊ - Chebyshev.psi ↑⌊t⌋₊ ≤ K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:253 < ⌊t⌋₊h_nat:↑⌊t⌋₊ - Chebyshev.psi ↑⌊t⌋₊ ≤ K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2hfl_le:↑⌊t⌋₊ ≤ t⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:253 < ⌊t⌋₊h_nat:↑⌊t⌋₊ - Chebyshev.psi ↑⌊t⌋₊ ≤ K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2hfl_le:↑⌊t⌋₊ ≤ thfl_pos:0 < ↑⌊t⌋₊⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:253 < ⌊t⌋₊h_nat:↑⌊t⌋₊ - Chebyshev.psi ↑⌊t⌋₊ ≤ K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2hfl_le:↑⌊t⌋₊ ≤ thfl_pos:0 < ↑⌊t⌋₊hlog_mono:Real.log ↑⌊t⌋₊ ≤ Real.log t⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:253 < ⌊t⌋₊h_nat:↑⌊t⌋₊ - Chebyshev.psi ↑⌊t⌋₊ ≤ K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2hfl_le:↑⌊t⌋₊ ≤ thfl_pos:0 < ↑⌊t⌋₊hlog_mono:Real.log ↑⌊t⌋₊ ≤ Real.log thlog_fl_nonneg:0 ≤ Real.log ↑⌊t⌋₊⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:253 < ⌊t⌋₊h_nat:↑⌊t⌋₊ - Chebyshev.psi ↑⌊t⌋₊ ≤ K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2hfl_le:↑⌊t⌋₊ ≤ thfl_pos:0 < ↑⌊t⌋₊hlog_mono:Real.log ↑⌊t⌋₊ ≤ Real.log thlog_fl_nonneg:0 ≤ Real.log ↑⌊t⌋₊hsq_mono:Real.log ↑⌊t⌋₊ ^ 2 ≤ Real.log t ^ 2⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:253 < ⌊t⌋₊h_nat:↑⌊t⌋₊ - Chebyshev.psi ↑⌊t⌋₊ ≤ K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2hfl_le:↑⌊t⌋₊ ≤ thfl_pos:0 < ↑⌊t⌋₊hlog_mono:Real.log ↑⌊t⌋₊ ≤ Real.log thlog_fl_nonneg:0 ≤ Real.log ↑⌊t⌋₊hsq_mono:Real.log ↑⌊t⌋₊ ^ 2 ≤ Real.log t ^ 2hsqrt_mono:√↑⌊t⌋₊ ≤ √t⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:253 < ⌊t⌋₊h_nat:↑⌊t⌋₊ - Chebyshev.psi ↑⌊t⌋₊ ≤ K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2hfl_le:↑⌊t⌋₊ ≤ thfl_pos:0 < ↑⌊t⌋₊hlog_mono:Real.log ↑⌊t⌋₊ ≤ Real.log thlog_fl_nonneg:0 ≤ Real.log ↑⌊t⌋₊hsq_mono:Real.log ↑⌊t⌋₊ ^ 2 ≤ Real.log t ^ 2hsqrt_mono:√↑⌊t⌋₊ ≤ √thlt_add1:t < ↑⌊t⌋₊ + 1⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:253 < ⌊t⌋₊h_nat:↑⌊t⌋₊ - Chebyshev.psi ↑⌊t⌋₊ ≤ K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2hfl_le:↑⌊t⌋₊ ≤ thfl_pos:0 < ↑⌊t⌋₊hlog_mono:Real.log ↑⌊t⌋₊ ≤ Real.log thlog_fl_nonneg:0 ≤ Real.log ↑⌊t⌋₊hsq_mono:Real.log ↑⌊t⌋₊ ^ 2 ≤ Real.log t ^ 2hsqrt_mono:√↑⌊t⌋₊ ≤ √thlt_add1:t < ↑⌊t⌋₊ + 1hmul_mono:K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2 ≤ K * √t * Real.log t ^ 2⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:253 < ⌊t⌋₊h_nat:↑⌊t⌋₊ - Chebyshev.psi ↑⌊t⌋₊ ≤ K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2hfl_le:↑⌊t⌋₊ ≤ thfl_pos:0 < ↑⌊t⌋₊hlog_mono:Real.log ↑⌊t⌋₊ ≤ Real.log thlog_fl_nonneg:0 ≤ Real.log ↑⌊t⌋₊hsq_mono:Real.log ↑⌊t⌋₊ ^ 2 ≤ Real.log t ^ 2hsqrt_mono:√↑⌊t⌋₊ ≤ √thlt_add1:t < ↑⌊t⌋₊ + 1hmul_mono:K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2 ≤ K * √t * Real.log t ^ 2hrpow_prod:√t * t ^ (α - 1 / 2) = t ^ α⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:253 < ⌊t⌋₊h_nat:↑⌊t⌋₊ - Chebyshev.psi ↑⌊t⌋₊ ≤ K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2hfl_le:↑⌊t⌋₊ ≤ thfl_pos:0 < ↑⌊t⌋₊hlog_mono:Real.log ↑⌊t⌋₊ ≤ Real.log thlog_fl_nonneg:0 ≤ Real.log ↑⌊t⌋₊hsq_mono:Real.log ↑⌊t⌋₊ ^ 2 ≤ Real.log t ^ 2hsqrt_mono:√↑⌊t⌋₊ ≤ √thlt_add1:t < ↑⌊t⌋₊ + 1hmul_mono:K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2 ≤ K * √t * Real.log t ^ 2hrpow_prod:√t * t ^ (α - 1 / 2) = t ^ αhmain:K * √t * Real.log t ^ 2 ≤ K * (1 / ε) ^ 2 * t ^ α⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:253 < ⌊t⌋₊h_nat:↑⌊t⌋₊ - Chebyshev.psi ↑⌊t⌋₊ ≤ K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2hfl_le:↑⌊t⌋₊ ≤ thfl_pos:0 < ↑⌊t⌋₊hlog_mono:Real.log ↑⌊t⌋₊ ≤ Real.log thlog_fl_nonneg:0 ≤ Real.log ↑⌊t⌋₊hsq_mono:Real.log ↑⌊t⌋₊ ^ 2 ≤ Real.log t ^ 2hsqrt_mono:√↑⌊t⌋₊ ≤ √thlt_add1:t < ↑⌊t⌋₊ + 1hmul_mono:K * √↑⌊t⌋₊ * Real.log ↑⌊t⌋₊ ^ 2 ≤ K * √t * Real.log t ^ 2hrpow_prod:√t * t ^ (α - 1 / 2) = t ^ αhmain:K * √t * Real.log t ^ 2 ≤ K * (1 / ε) ^ 2 * t ^ αhpsi_eq:Chebyshev.psi t = Chebyshev.psi ↑⌊t⌋₊⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
All goals completed! 🐙
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:¬253 < ⌊t⌋₊⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:⌊t⌋₊ ≤ 253⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:⌊t⌋₊ ≤ 253hfl_le:↑⌊t⌋₊ ≤ 253⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:⌊t⌋₊ ≤ 253hfl_le:↑⌊t⌋₊ ≤ 253hlt_254:t < 254⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:⌊t⌋₊ ≤ 253hfl_le:↑⌊t⌋₊ ≤ 253hlt_254:t < 254hpsi_nonneg:0 ≤ Chebyshev.psi t⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
K:ℝhK:0 < Kh:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2α:ℝhα:1 / 2 < αε:ℝ := (α - 1 / 2) / 2hε:0 < εt:ℝht:1 ≤ tht0:0 ≤ tht_pos:0 < thlog_le:Real.log t ≤ t ^ ε / εhlog_nonneg:0 ≤ Real.log thsq:Real.log t ^ 2 ≤ (1 / ε) ^ 2 * t ^ (α - 1 / 2)hrpow_ge_one:1 ≤ t ^ αhfloor:⌊t⌋₊ ≤ 253hfl_le:↑⌊t⌋₊ ≤ 253hlt_254:t < 254hpsi_nonneg:0 ≤ Chebyshev.psi thpos_term:0 ≤ K * (1 / ε) ^ 2 * t ^ α⊢ t - Chebyshev.psi t ≤ (K * (1 / ε) ^ 2 + 255) * t ^ α
All goals completed! 🐙
Residue contradiction at any zero ρ (α < ρ.re < 1) when psiNegDefect C α ≥ 0.
lemma not_riemannZeta_eq_zero_of_psiNegDefect_nonneg {C α : ℝ} {ρ : ℂ}
(hα_pos : 0 < α) (hα_lt_ρ : α < ρ.re) (hρ_lt : ρ.re < 1)
(hg_nonneg : ∀ t : ℝ, 1 ≤ t → 0 ≤ psiNegDefect C α t) :
riemannZeta ρ ≠ 0 := C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α t⊢ riemannZeta ρ ≠ 0
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volume⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρ⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρh_lhs_tendsto:Tendsto (fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) (𝓝[≠] ρ) (𝓝 0)⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρh_lhs_tendsto:Tendsto (fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) (𝓝[≠] ρ) (𝓝 0)hρ1:ρ ≠ 1⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρh_lhs_tendsto:Tendsto (fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) (𝓝[≠] ρ) (𝓝 0)hρ1:ρ ≠ 1hρ0:ρ ≠ 0⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρh_lhs_tendsto:Tendsto (fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) (𝓝[≠] ρ) (𝓝 0)hρ1:ρ ≠ 1hρ0:ρ ≠ 0m:ℕ := analyticOrderNatAt riemannZeta ρ⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρh_lhs_tendsto:Tendsto (fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) (𝓝[≠] ρ) (𝓝 0)hρ1:ρ ≠ 1hρ0:ρ ≠ 0m:ℕ := analyticOrderNatAt riemannZeta ρhm_ge_one:1 ≤ m⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρh_lhs_tendsto:Tendsto (fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) (𝓝[≠] ρ) (𝓝 0)hρ1:ρ ≠ 1hρ0:ρ ≠ 0m:ℕ := analyticOrderNatAt riemannZeta ρhm_ge_one:1 ≤ mhm_ne_zero:↑m ≠ 0⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρh_lhs_tendsto:Tendsto (fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) (𝓝[≠] ρ) (𝓝 0)hρ1:ρ ≠ 1hρ0:ρ ≠ 0m:ℕ := analyticOrderNatAt riemannZeta ρhm_ge_one:1 ≤ mhm_ne_zero:↑m ≠ 0h_res:Tendsto (fun z ↦ (z - ρ) * (deriv riemannZeta z / riemannZeta z)) (𝓝[≠] ρ) (𝓝 ↑(analyticOrderNatAt riemannZeta ρ))⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρh_lhs_tendsto:Tendsto (fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) (𝓝[≠] ρ) (𝓝 0)hρ1:ρ ≠ 1hρ0:ρ ≠ 0m:ℕ := analyticOrderNatAt riemannZeta ρhm_ge_one:1 ≤ mhm_ne_zero:↑m ≠ 0h_res:Tendsto (fun z ↦ (z - ρ) * (deriv riemannZeta z / riemannZeta z)) (𝓝[≠] ρ) (𝓝 ↑(analyticOrderNatAt riemannZeta ρ))h_zeta_ne:∀ᶠ (z : ℂ) in 𝓝[≠] ρ, riemannZeta z ≠ 0⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρh_lhs_tendsto:Tendsto (fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) (𝓝[≠] ρ) (𝓝 0)hρ1:ρ ≠ 1hρ0:ρ ≠ 0m:ℕ := analyticOrderNatAt riemannZeta ρhm_ge_one:1 ≤ mhm_ne_zero:↑m ≠ 0h_res:Tendsto (fun z ↦ (z - ρ) * (deriv riemannZeta z / riemannZeta z)) (𝓝[≠] ρ) (𝓝 ↑(analyticOrderNatAt riemannZeta ρ))h_zeta_ne:∀ᶠ (z : ℂ) in 𝓝[≠] ρ, riemannZeta z ≠ 0h_open_nhds:∀ᶠ (z : ℂ) in 𝓝[≠] ρ, α < z.re ∧ z ≠ 1 ∧ z ≠ 0⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρh_lhs_tendsto:Tendsto (fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) (𝓝[≠] ρ) (𝓝 0)hρ1:ρ ≠ 1hρ0:ρ ≠ 0m:ℕ := analyticOrderNatAt riemannZeta ρhm_ge_one:1 ≤ mhm_ne_zero:↑m ≠ 0h_res:Tendsto (fun z ↦ (z - ρ) * (deriv riemannZeta z / riemannZeta z)) (𝓝[≠] ρ) (𝓝 ↑(analyticOrderNatAt riemannZeta ρ))h_zeta_ne:∀ᶠ (z : ℂ) in 𝓝[≠] ρ, riemannZeta z ≠ 0h_open_nhds:∀ᶠ (z : ℂ) in 𝓝[≠] ρ, α < z.re ∧ z ≠ 1 ∧ z ≠ 0h_congr:(fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) =ᶠ[𝓝[≠] ρ] fun z ↦
(z - ρ) * (↑C / (z - ↑α) - 1 / (z - 1)) - (z - ρ) * (deriv riemannZeta z / riemannZeta z) / z⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρh_lhs_tendsto:Tendsto (fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) (𝓝[≠] ρ) (𝓝 0)hρ1:ρ ≠ 1hρ0:ρ ≠ 0m:ℕ := analyticOrderNatAt riemannZeta ρhm_ge_one:1 ≤ mhm_ne_zero:↑m ≠ 0h_res:Tendsto (fun z ↦ (z - ρ) * (deriv riemannZeta z / riemannZeta z)) (𝓝[≠] ρ) (𝓝 ↑(analyticOrderNatAt riemannZeta ρ))h_zeta_ne:∀ᶠ (z : ℂ) in 𝓝[≠] ρ, riemannZeta z ≠ 0h_open_nhds:∀ᶠ (z : ℂ) in 𝓝[≠] ρ, α < z.re ∧ z ≠ 1 ∧ z ≠ 0h_congr:(fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) =ᶠ[𝓝[≠] ρ] fun z ↦
(z - ρ) * (↑C / (z - ↑α) - 1 / (z - 1)) - (z - ρ) * (deriv riemannZeta z / riemannZeta z) / zh_ρ_sub_α:ρ - ↑α ≠ 0⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρh_lhs_tendsto:Tendsto (fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) (𝓝[≠] ρ) (𝓝 0)hρ1:ρ ≠ 1hρ0:ρ ≠ 0m:ℕ := analyticOrderNatAt riemannZeta ρhm_ge_one:1 ≤ mhm_ne_zero:↑m ≠ 0h_res:Tendsto (fun z ↦ (z - ρ) * (deriv riemannZeta z / riemannZeta z)) (𝓝[≠] ρ) (𝓝 ↑(analyticOrderNatAt riemannZeta ρ))h_zeta_ne:∀ᶠ (z : ℂ) in 𝓝[≠] ρ, riemannZeta z ≠ 0h_open_nhds:∀ᶠ (z : ℂ) in 𝓝[≠] ρ, α < z.re ∧ z ≠ 1 ∧ z ≠ 0h_congr:(fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) =ᶠ[𝓝[≠] ρ] fun z ↦
(z - ρ) * (↑C / (z - ↑α) - 1 / (z - 1)) - (z - ρ) * (deriv riemannZeta z / riemannZeta z) / zh_ρ_sub_α:ρ - ↑α ≠ 0h_first_tendsto:Tendsto (fun z ↦ (z - ρ) * (↑C / (z - ↑α) - 1 / (z - 1))) (𝓝[≠] ρ) (𝓝 0)⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρh_lhs_tendsto:Tendsto (fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) (𝓝[≠] ρ) (𝓝 0)hρ1:ρ ≠ 1hρ0:ρ ≠ 0m:ℕ := analyticOrderNatAt riemannZeta ρhm_ge_one:1 ≤ mhm_ne_zero:↑m ≠ 0h_res:Tendsto (fun z ↦ (z - ρ) * (deriv riemannZeta z / riemannZeta z)) (𝓝[≠] ρ) (𝓝 ↑(analyticOrderNatAt riemannZeta ρ))h_zeta_ne:∀ᶠ (z : ℂ) in 𝓝[≠] ρ, riemannZeta z ≠ 0h_open_nhds:∀ᶠ (z : ℂ) in 𝓝[≠] ρ, α < z.re ∧ z ≠ 1 ∧ z ≠ 0h_congr:(fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) =ᶠ[𝓝[≠] ρ] fun z ↦
(z - ρ) * (↑C / (z - ↑α) - 1 / (z - 1)) - (z - ρ) * (deriv riemannZeta z / riemannZeta z) / zh_ρ_sub_α:ρ - ↑α ≠ 0h_first_tendsto:Tendsto (fun z ↦ (z - ρ) * (↑C / (z - ↑α) - 1 / (z - 1))) (𝓝[≠] ρ) (𝓝 0)h_rhs_tendsto:Tendsto (fun z ↦ (z - ρ) * (↑C / (z - ↑α) - 1 / (z - 1)) - (z - ρ) * (deriv riemannZeta z / riemannZeta z) / z) (𝓝[≠] ρ)
(𝓝 (-↑m / ρ))⊢ False
C:ℝα:ℝρ:ℂhα_pos:0 < αhα_lt_ρ:α < ρ.rehρ_lt:ρ.re < 1hg_nonneg:∀ (t : ℝ), 1 ≤ t → 0 ≤ psiNegDefect C α thρ_zero:riemannZeta ρ = 0hα_lt_one:α < 1hInt_α:∀ (σ : ℝ), α < σ → IntegrableOn (fun t ↦ psiNegDefect C α t * t ^ (-(σ + 1))) (Set.Ioi 1) volumeh_cont_mellin:ContinuousAt (mellinLogDeriv (psiNegDefect C α) 0) ρh_lhs_tendsto:Tendsto (fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) (𝓝[≠] ρ) (𝓝 0)hρ1:ρ ≠ 1hρ0:ρ ≠ 0m:ℕ := analyticOrderNatAt riemannZeta ρhm_ge_one:1 ≤ mhm_ne_zero:↑m ≠ 0h_res:Tendsto (fun z ↦ (z - ρ) * (deriv riemannZeta z / riemannZeta z)) (𝓝[≠] ρ) (𝓝 ↑(analyticOrderNatAt riemannZeta ρ))h_zeta_ne:∀ᶠ (z : ℂ) in 𝓝[≠] ρ, riemannZeta z ≠ 0h_open_nhds:∀ᶠ (z : ℂ) in 𝓝[≠] ρ, α < z.re ∧ z ≠ 1 ∧ z ≠ 0h_congr:(fun z ↦ (z - ρ) * mellinLogDeriv (psiNegDefect C α) 0 z) =ᶠ[𝓝[≠] ρ] fun z ↦
(z - ρ) * (↑C / (z - ↑α) - 1 / (z - 1)) - (z - ρ) * (deriv riemannZeta z / riemannZeta z) / zh_ρ_sub_α:ρ - ↑α ≠ 0h_first_tendsto:Tendsto (fun z ↦ (z - ρ) * (↑C / (z - ↑α) - 1 / (z - 1))) (𝓝[≠] ρ) (𝓝 0)h_rhs_tendsto:Tendsto (fun z ↦ (z - ρ) * (↑C / (z - ↑α) - 1 / (z - 1)) - (z - ρ) * (deriv riemannZeta z / riemannZeta z) / z) (𝓝[≠] ρ)
(𝓝 (-↑m / ρ))h_eq_lim:0 = -↑m / ρ⊢ False
All goals completed! 🐙
One-Sided Landau Oscillation Theorem (t - ψ(t)):
If ¬ RiemannHypothesis, then for every constant K > 0, there exists x : ℕ with 253 < x
and K * Real.sqrt x * (Real.log x) ^ 2 < (x : ℝ) - Chebyshev.psi x.
theorem not_riemannHypothesis_imp_exists_self_sub_psi_gt
(hNotRH : ¬ RiemannHypothesis) {K : ℝ} (hK : 0 < K) :
∃ x : ℕ, 253 < x ∧ K * Real.sqrt x * (Real.log x) ^ 2 < (x : ℝ) - Chebyshev.psi x := hNotRH:¬RiemannHypothesisK:ℝhK:0 < K⊢ ∃ x, 253 < x ∧ K * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑x
hNotRH:¬RiemannHypothesisK:ℝhK:0 < Kh_no:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2⊢ False
hNotRH:¬RiemannHypothesisK:ℝhK:0 < Kh_no:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2ρ:ℂhρ_zero:riemannZeta ρ = 0hρ_gt:1 / 2 < ρ.rehρ_lt:ρ.re < 1⊢ False
hNotRH:¬RiemannHypothesisK:ℝhK:0 < Kh_no:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2ρ:ℂhρ_zero:riemannZeta ρ = 0hρ_gt:1 / 2 < ρ.rehρ_lt:ρ.re < 1α:ℝ := (1 / 2 + ρ.re) / 2hα:α = (1 / 2 + ρ.re) / 2⊢ False
hNotRH:¬RiemannHypothesisK:ℝhK:0 < Kh_no:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2ρ:ℂhρ_zero:riemannZeta ρ = 0hρ_gt:1 / 2 < ρ.rehρ_lt:ρ.re < 1α:ℝ := (1 / 2 + ρ.re) / 2hα:α = (1 / 2 + ρ.re) / 2C:ℝ_hC_pos:0 < Ch_bound:∀ (t : ℝ), 1 ≤ t → t - Chebyshev.psi t ≤ C * t ^ α⊢ False
exact not_riemannZeta_eq_zero_of_psiNegDefect_nonneg (C := C) (α := α)
(hNotRH:¬RiemannHypothesisK:ℝhK:0 < Kh_no:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2ρ:ℂhρ_zero:riemannZeta ρ = 0hρ_gt:1 / 2 < ρ.rehρ_lt:ρ.re < 1α:ℝ := (1 / 2 + ρ.re) / 2hα:α = (1 / 2 + ρ.re) / 2C:ℝ_hC_pos:0 < Ch_bound:∀ (t : ℝ), 1 ≤ t → t - Chebyshev.psi t ≤ C * t ^ α⊢ 0 < α All goals completed! 🐙) (hNotRH:¬RiemannHypothesisK:ℝhK:0 < Kh_no:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2ρ:ℂhρ_zero:riemannZeta ρ = 0hρ_gt:1 / 2 < ρ.rehρ_lt:ρ.re < 1α:ℝ := (1 / 2 + ρ.re) / 2hα:α = (1 / 2 + ρ.re) / 2C:ℝ_hC_pos:0 < Ch_bound:∀ (t : ℝ), 1 ≤ t → t - Chebyshev.psi t ≤ C * t ^ α⊢ α < ρ.re All goals completed! 🐙) hρ_lt
(fun t ht ↦ hNotRH:¬RiemannHypothesisK:ℝhK:0 < Kh_no:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2ρ:ℂhρ_zero:riemannZeta ρ = 0hρ_gt:1 / 2 < ρ.rehρ_lt:ρ.re < 1α:ℝ := (1 / 2 + ρ.re) / 2hα:α = (1 / 2 + ρ.re) / 2C:ℝ_hC_pos:0 < Ch_bound:∀ (t : ℝ), 1 ≤ t → t - Chebyshev.psi t ≤ C * t ^ αt:ℝht:1 ≤ t⊢ 0 ≤ psiNegDefect C α t hNotRH:¬RiemannHypothesisK:ℝhK:0 < Kh_no:∀ (x : ℕ), 253 < x → ↑x - Chebyshev.psi ↑x ≤ K * √↑x * Real.log ↑x ^ 2ρ:ℂhρ_zero:riemannZeta ρ = 0hρ_gt:1 / 2 < ρ.rehρ_lt:ρ.re < 1α:ℝ := (1 / 2 + ρ.re) / 2hα:α = (1 / 2 + ρ.re) / 2C:ℝ_hC_pos:0 < Ch_bound:∀ (t : ℝ), 1 ≤ t → t - Chebyshev.psi t ≤ C * t ^ αt:ℝht:1 ≤ t⊢ 0 ≤ C * t ^ α + Chebyshev.psi t - t; All goals completed! 🐙) hρ_zero
If x > 253 satisfies 10 * √x * (log x)^2 < x - ψ(x), then ShuniaCounterexample x holds.
theorem shuniaCounterexample_of_self_sub_psi_gt {x : ℕ} (hx : 253 < x)
(hpsi : 10 * Real.sqrt x * (Real.log x) ^ 2 < (x : ℝ) - Chebyshev.psi x) :
ShuniaCounterexample x := x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑x⊢ ShuniaCounterexample x
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xhno:¬ShuniaCounterexample x⊢ False
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xhno:¬(quarterClock (x - 1) + 1) * 2 ^ quarterClock (x - 1) ≤ shuniaDeficit x⊢ False
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xhno:shuniaDeficit x < (quarterClock (x - 1) + 1) * 2 ^ quarterClock (x - 1)⊢ False
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xq:ℕ := quarterClock (x - 1)hno:shuniaDeficit x < (q + 1) * 2 ^ q⊢ False
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xq:ℕ := quarterClock (x - 1)hno:shuniaDeficit x < (q + 1) * 2 ^ qhx2:2 ≤ x⊢ False
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xq:ℕ := quarterClock (x - 1)hno:shuniaDeficit x < (q + 1) * 2 ^ qhx2:2 ≤ xhlo:4 ^ quarterClock (x - 1) ≤ 3 * xright✝:3 * x < 4 ^ (quarterClock (x - 1) + 1)⊢ False
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xq:ℕ := quarterClock (x - 1)hno:shuniaDeficit x < (q + 1) * 2 ^ qhx2:2 ≤ xhlo:4 ^ quarterClock (x - 1) ≤ 3 * xright✝:3 * x < 4 ^ (quarterClock (x - 1) + 1)h2q_sq:↑(2 ^ q) ^ 2 ≤ 3 * ↑x⊢ False
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xq:ℕ := quarterClock (x - 1)hno:shuniaDeficit x < (q + 1) * 2 ^ qhx2:2 ≤ xhlo:4 ^ quarterClock (x - 1) ≤ 3 * xright✝:3 * x < 4 ^ (quarterClock (x - 1) + 1)h2q_sq:↑(2 ^ q) ^ 2 ≤ 3 * ↑xh2q_le:↑(2 ^ q) ≤ 2 * √↑x⊢ False
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xq:ℕ := quarterClock (x - 1)hno:shuniaDeficit x < (q + 1) * 2 ^ qhx2:2 ≤ xhlo:4 ^ quarterClock (x - 1) ≤ 3 * xright✝:3 * x < 4 ^ (quarterClock (x - 1) + 1)h2q_sq:↑(2 ^ q) ^ 2 ≤ 3 * ↑xh2q_le:↑(2 ^ q) ≤ 2 * √↑xhq_log:↑q ≤ 3 * Real.log ↑x⊢ False
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xq:ℕ := quarterClock (x - 1)hno:shuniaDeficit x < (q + 1) * 2 ^ qhx2:2 ≤ xhlo:4 ^ quarterClock (x - 1) ≤ 3 * xright✝:3 * x < 4 ^ (quarterClock (x - 1) + 1)h2q_sq:↑(2 ^ q) ^ 2 ≤ 3 * ↑xh2q_le:↑(2 ^ q) ≤ 2 * √↑xhq_log:↑q ≤ 3 * Real.log ↑xhlog_pos:0 < Real.log ↑x⊢ False
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xq:ℕ := quarterClock (x - 1)hno:shuniaDeficit x < (q + 1) * 2 ^ qhx2:2 ≤ xhlo:4 ^ quarterClock (x - 1) ≤ 3 * xright✝:3 * x < 4 ^ (quarterClock (x - 1) + 1)h2q_sq:↑(2 ^ q) ^ 2 ≤ 3 * ↑xh2q_le:↑(2 ^ q) ≤ 2 * √↑xhq_log:↑q ≤ 3 * Real.log ↑xhlog_pos:0 < Real.log ↑xhlog_ge5:5 ≤ Real.log ↑x⊢ False
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xq:ℕ := quarterClock (x - 1)hno:shuniaDeficit x < (q + 1) * 2 ^ qhx2:2 ≤ xhlo:4 ^ quarterClock (x - 1) ≤ 3 * xright✝:3 * x < 4 ^ (quarterClock (x - 1) + 1)h2q_sq:↑(2 ^ q) ^ 2 ≤ 3 * ↑xh2q_le:↑(2 ^ q) ≤ 2 * √↑xhq_log:↑q ≤ 3 * Real.log ↑xhlog_pos:0 < Real.log ↑xhlog_ge5:5 ≤ Real.log ↑xhsqrt_ge15:15 ≤ √↑x⊢ False
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xq:ℕ := quarterClock (x - 1)hno:shuniaDeficit x < (q + 1) * 2 ^ qhx2:2 ≤ xhlo:4 ^ quarterClock (x - 1) ≤ 3 * xright✝:3 * x < 4 ^ (quarterClock (x - 1) + 1)h2q_sq:↑(2 ^ q) ^ 2 ≤ 3 * ↑xh2q_le:↑(2 ^ q) ≤ 2 * √↑xhq_log:↑q ≤ 3 * Real.log ↑xhlog_pos:0 < Real.log ↑xhlog_ge5:5 ≤ Real.log ↑xhsqrt_ge15:15 ≤ √↑xhdef_le:↑(shuniaDeficit x) ≤ 8 * √↑x * Real.log ↑x⊢ False
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xq:ℕ := quarterClock (x - 1)hno:shuniaDeficit x < (q + 1) * 2 ^ qhx2:2 ≤ xhlo:4 ^ quarterClock (x - 1) ≤ 3 * xright✝:3 * x < 4 ^ (quarterClock (x - 1) + 1)h2q_sq:↑(2 ^ q) ^ 2 ≤ 3 * ↑xh2q_le:↑(2 ^ q) ≤ 2 * √↑xhq_log:↑q ≤ 3 * Real.log ↑xhlog_pos:0 < Real.log ↑xhlog_ge5:5 ≤ Real.log ↑xhsqrt_ge15:15 ≤ √↑xhdef_le:↑(shuniaDeficit x) ≤ 8 * √↑x * Real.log ↑xhtheta_le:↑x - Chebyshev.theta ↑x ≤ (↑(shuniaDeficit x) + 3) * Real.log ↑x + 1⊢ False
x:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑xq:ℕ := quarterClock (x - 1)hno:shuniaDeficit x < (q + 1) * 2 ^ qhx2:2 ≤ xhlo:4 ^ quarterClock (x - 1) ≤ 3 * xright✝:3 * x < 4 ^ (quarterClock (x - 1) + 1)h2q_sq:↑(2 ^ q) ^ 2 ≤ 3 * ↑xh2q_le:↑(2 ^ q) ≤ 2 * √↑xhq_log:↑q ≤ 3 * Real.log ↑xhlog_pos:0 < Real.log ↑xhlog_ge5:5 ≤ Real.log ↑xhsqrt_ge15:15 ≤ √↑xhdef_le:↑(shuniaDeficit x) ≤ 8 * √↑x * Real.log ↑xhtheta_le:↑x - Chebyshev.theta ↑x ≤ (↑(shuniaDeficit x) + 3) * Real.log ↑x + 1hpsi_ge_theta:Chebyshev.theta ↑x ≤ Chebyshev.psi ↑x⊢ False
All goals completed! 🐙
One-Sided Landau Converse (¬ RiemannHypothesis → ∃ n ≥ 2, ShuniaCounterexample n):
If RiemannHypothesis is false, there exists n ≥ 2 satisfying ShuniaCounterexample n.
theorem not_riemannHypothesis_imp_exists_shuniaCounterexample
(hNotRH : ¬ RiemannHypothesis) :
∃ n ≥ 2, ShuniaCounterexample n := hNotRH:¬RiemannHypothesis⊢ ∃ n ≥ 2, ShuniaCounterexample n
hNotRH:¬RiemannHypothesisx:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑x⊢ ∃ n ≥ 2, ShuniaCounterexample n
exact ⟨x, hNotRH:¬RiemannHypothesisx:ℕhx:253 < xhpsi:10 * √↑x * Real.log ↑x ^ 2 < ↑x - Chebyshev.psi ↑x⊢ x ≥ 2 All goals completed! 🐙, shuniaCounterexample_of_self_sub_psi_gt hx hpsi⟩5. Final Equivalence Bridge
If (n : ℝ) - θ(n) < (q_n + 1) * 2 ^ q_n * log n for n ≥ 2, then
¬ ShuniaCounterexample n.
theorem not_shuniaCounterexample_of_theta_bound {n : ℕ} (hn2 : 2 ≤ n)
(hbound : (n : ℝ) - Chebyshev.theta n <
((quarterClock (n - 1) + 1) * 2 ^ quarterClock (n - 1) : ℝ) * Real.log n) :
¬ ShuniaCounterexample n := n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑n⊢ ¬ShuniaCounterexample n
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑nhce:ShuniaCounterexample n⊢ False
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑nhce:ShuniaCounterexample nhlog_pos:0 < Real.log ↑n⊢ False
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑nhce:ShuniaCounterexample nhlog_pos:0 < Real.log ↑nhce_r:↑((quarterClock (n - 1) + 1) * 2 ^ quarterClock (n - 1)) * Real.log ↑n ≤ ↑(shuniaDeficit n) * Real.log ↑n⊢ False
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑nhce:ShuniaCounterexample nhlog_pos:0 < Real.log ↑nhce_r:↑((quarterClock (n - 1) + 1) * 2 ^ quarterClock (n - 1)) * Real.log ↑n ≤ ↑(shuniaDeficit n) * Real.log ↑nhdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1⊢ False
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑nhce:ShuniaCounterexample nhlog_pos:0 < Real.log ↑nhdef:↑(shuniaDeficit n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1hce_r:(↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑n ≤ ↑(shuniaDeficit n) * Real.log ↑n⊢ False
All goals completed! 🐙
Schoenfeld's explicit bound (n : ℝ) - θ(n) < √n * (log n)^2 / (8 * π) (or even
≤ (1 / 2) * √n * (log n)^2) implies the large-n quartering threshold for all n ≥ 2.
theorem large_theta_bound_of_half_sqrt_log_sq {n : ℕ} (hn2 : 2 ≤ n)
(hbound : (n : ℝ) - Chebyshev.theta n ≤ (1 / 2 : ℝ) * Real.sqrt n * (Real.log n) ^ 2) :
(n : ℝ) - Chebyshev.theta n <
((quarterClock (n - 1) + 1) * 2 ^ quarterClock (n - 1) : ℝ) * Real.log n := n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n ≤ 1 / 2 * √↑n * Real.log ↑n ^ 2⊢ ↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑n
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n ≤ 1 / 2 * √↑n * Real.log ↑n ^ 2q:ℕ := quarterClock (n - 1)⊢ ↑n - Chebyshev.theta ↑n < (↑q + 1) * 2 ^ q * Real.log ↑n
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n ≤ 1 / 2 * √↑n * Real.log ↑n ^ 2q:ℕ := quarterClock (n - 1)left✝:4 ^ quarterClock (n - 1) ≤ 3 * nhhi:3 * n < 4 ^ (quarterClock (n - 1) + 1)⊢ ↑n - Chebyshev.theta ↑n < (↑q + 1) * 2 ^ q * Real.log ↑n
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n ≤ 1 / 2 * √↑n * Real.log ↑n ^ 2q:ℕ := quarterClock (n - 1)left✝:4 ^ quarterClock (n - 1) ≤ 3 * nhhi:3 * n < 4 ^ (quarterClock (n - 1) + 1)hlog_pos:0 < Real.log ↑n⊢ ↑n - Chebyshev.theta ↑n < (↑q + 1) * 2 ^ q * Real.log ↑n
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n ≤ 1 / 2 * √↑n * Real.log ↑n ^ 2q:ℕ := quarterClock (n - 1)left✝:4 ^ quarterClock (n - 1) ≤ 3 * nhhi:3 * n < 4 ^ (quarterClock (n - 1) + 1)hlog_pos:0 < Real.log ↑nhsqrt_lt:√↑n < ↑(2 ^ (q + 1)) / √3⊢ ↑n - Chebyshev.theta ↑n < (↑q + 1) * 2 ^ q * Real.log ↑n
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n ≤ 1 / 2 * √↑n * Real.log ↑n ^ 2q:ℕ := quarterClock (n - 1)left✝:4 ^ quarterClock (n - 1) ≤ 3 * nhhi:3 * n < 4 ^ (quarterClock (n - 1) + 1)hlog_pos:0 < Real.log ↑nhsqrt_lt:√↑n < ↑(2 ^ (q + 1)) / √3hsqrt3_gt:17 / 10 < √3⊢ ↑n - Chebyshev.theta ↑n < (↑q + 1) * 2 ^ q * Real.log ↑n
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n ≤ 1 / 2 * √↑n * Real.log ↑n ^ 2q:ℕ := quarterClock (n - 1)left✝:4 ^ quarterClock (n - 1) ≤ 3 * nhhi:3 * n < 4 ^ (quarterClock (n - 1) + 1)hlog_pos:0 < Real.log ↑nhsqrt_lt:√↑n < ↑(2 ^ (q + 1)) / √3hsqrt3_gt:17 / 10 < √3hsqrt_le:√↑n ≤ 20 / 17 * ↑(2 ^ q)⊢ ↑n - Chebyshev.theta ↑n < (↑q + 1) * 2 ^ q * Real.log ↑n
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n ≤ 1 / 2 * √↑n * Real.log ↑n ^ 2q:ℕ := quarterClock (n - 1)left✝:4 ^ quarterClock (n - 1) ≤ 3 * nhhi:3 * n < 4 ^ (quarterClock (n - 1) + 1)hlog_pos:0 < Real.log ↑nhsqrt_lt:√↑n < ↑(2 ^ (q + 1)) / √3hsqrt3_gt:17 / 10 < √3hsqrt_le:√↑n ≤ 20 / 17 * ↑(2 ^ q)hlog_lt:Real.log ↑n < ↑(q + 1) * (7 / 5)⊢ ↑n - Chebyshev.theta ↑n < (↑q + 1) * 2 ^ q * Real.log ↑n
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n ≤ 1 / 2 * √↑n * Real.log ↑n ^ 2q:ℕ := quarterClock (n - 1)left✝:4 ^ quarterClock (n - 1) ≤ 3 * nhhi:3 * n < 4 ^ (quarterClock (n - 1) + 1)hlog_pos:0 < Real.log ↑nhsqrt_lt:√↑n < ↑(2 ^ (q + 1)) / √3hsqrt3_gt:17 / 10 < √3hsqrt_le:√↑n ≤ 20 / 17 * ↑(2 ^ q)hlog_lt:Real.log ↑n < ↑(q + 1) * (7 / 5)hprod:1 / 2 * √↑n * Real.log ↑n < 1 / 2 * (20 / 17 * ↑(2 ^ q)) * (↑(q + 1) * (7 / 5))⊢ ↑n - Chebyshev.theta ↑n < (↑q + 1) * 2 ^ q * Real.log ↑n
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n ≤ 1 / 2 * √↑n * Real.log ↑n ^ 2q:ℕ := quarterClock (n - 1)left✝:4 ^ quarterClock (n - 1) ≤ 3 * nhhi:3 * n < 4 ^ (quarterClock (n - 1) + 1)hlog_pos:0 < Real.log ↑nhsqrt_lt:√↑n < ↑(2 ^ (q + 1)) / √3hsqrt3_gt:17 / 10 < √3hsqrt_le:√↑n ≤ 20 / 17 * ↑(2 ^ q)hlog_lt:Real.log ↑n < ↑(q + 1) * (7 / 5)hprod:1 / 2 * √↑n * Real.log ↑n < 1 / 2 * (20 / 17 * ↑(2 ^ q)) * (↑(q + 1) * (7 / 5))hcoeff:1 / 2 * (20 / 17 * ↑(2 ^ q)) * (↑(q + 1) * (7 / 5)) = 14 / 17 * ((↑q + 1) * 2 ^ q)⊢ ↑n - Chebyshev.theta ↑n < (↑q + 1) * 2 ^ q * Real.log ↑n
n:ℕhn2:2 ≤ nhbound:↑n - Chebyshev.theta ↑n ≤ 1 / 2 * √↑n * Real.log ↑n ^ 2q:ℕ := quarterClock (n - 1)left✝:4 ^ quarterClock (n - 1) ≤ 3 * nhhi:3 * n < 4 ^ (quarterClock (n - 1) + 1)hlog_pos:0 < Real.log ↑nhsqrt_lt:√↑n < ↑(2 ^ (q + 1)) / √3hsqrt3_gt:17 / 10 < √3hsqrt_le:√↑n ≤ 20 / 17 * ↑(2 ^ q)hlog_lt:Real.log ↑n < ↑(q + 1) * (7 / 5)hprod:1 / 2 * √↑n * Real.log ↑n < 1 / 2 * (20 / 17 * ↑(2 ^ q)) * (↑(q + 1) * (7 / 5))hcoeff:1 / 2 * (20 / 17 * ↑(2 ^ q)) * (↑(q + 1) * (7 / 5)) = 14 / 17 * ((↑q + 1) * 2 ^ q)hpos_rhs:0 < (↑q + 1) * 2 ^ q⊢ ↑n - Chebyshev.theta ↑n < (↑q + 1) * 2 ^ q * Real.log ↑n
All goals completed! 🐙
Main Bridge Theorem (ShuniaCounterexample ↔ ¬ RiemannHypothesis):
Given the large-n RH bound (n ≥ 21846), there exists n ≥ 2 satisfying
ShuniaCounterexample n if and only if ¬ RiemannHypothesis.
theorem exists_shuniaCounterexample_iff_not_riemannHypothesis_of_large
(hLarge : RiemannHypothesis → ∀ n : ℕ, 21846 ≤ n →
(n : ℝ) - Chebyshev.theta n <
((quarterClock (n - 1) + 1) * 2 ^ quarterClock (n - 1) : ℝ) * Real.log n) :
(∃ n ≥ 2, ShuniaCounterexample n) ↔ ¬ RiemannHypothesis := hLarge:RiemannHypothesis →
∀ (n : ℕ),
21846 ≤ n → ↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑n⊢ (∃ n ≥ 2, ShuniaCounterexample n) ↔ ¬RiemannHypothesis
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21846 ≤ n → ↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑n⊢ (∃ n ≥ 2, ShuniaCounterexample n) → ¬RiemannHypothesishLarge:RiemannHypothesis →
∀ (n : ℕ),
21846 ≤ n → ↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑n⊢ ¬RiemannHypothesis → ∃ n ≥ 2, ShuniaCounterexample n
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21846 ≤ n → ↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑n⊢ (∃ n ≥ 2, ShuniaCounterexample n) → ¬RiemannHypothesis hLarge:RiemannHypothesis →
∀ (n : ℕ),
21846 ≤ n → ↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑nn:ℕhn2:n ≥ 2hce:ShuniaCounterexample nhRH:RiemannHypothesis⊢ False
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21846 ≤ n → ↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑nn:ℕhn2:n ≥ 2hce:ShuniaCounterexample nhRH:RiemannHypothesishsmall:n ≤ 21845⊢ FalsehLarge:RiemannHypothesis →
∀ (n : ℕ),
21846 ≤ n → ↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑nn:ℕhn2:n ≥ 2hce:ShuniaCounterexample nhRH:RiemannHypothesishlarge:21845 < n⊢ False
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21846 ≤ n → ↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑nn:ℕhn2:n ≥ 2hce:ShuniaCounterexample nhRH:RiemannHypothesishsmall:n ≤ 21845⊢ False All goals completed! 🐙
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21846 ≤ n → ↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑nn:ℕhn2:n ≥ 2hce:ShuniaCounterexample nhRH:RiemannHypothesishlarge:21845 < n⊢ False exact not_shuniaCounterexample_of_theta_bound hn2 (hLarge hRH n (hLarge:RiemannHypothesis →
∀ (n : ℕ),
21846 ≤ n → ↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑nn:ℕhn2:n ≥ 2hce:ShuniaCounterexample nhRH:RiemannHypothesishlarge:21845 < n⊢ 21846 ≤ n All goals completed! 🐙)) hce
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21846 ≤ n → ↑n - Chebyshev.theta ↑n < (↑(quarterClock (n - 1)) + 1) * 2 ^ quarterClock (n - 1) * Real.log ↑n⊢ ¬RiemannHypothesis → ∃ n ≥ 2, ShuniaCounterexample n All goals completed! 🐙5. Exact 5-Register Machine Alignment Exports
Exact prime-product register value P(n) = ∏_{1 ≤ p ≤ n, p prime} p.
def shuniaPrimorial (n : ℕ) : ℕ :=
((Finset.Icc 1 n).filter Nat.Prime).prod id
Exact base-n offset-digit deficit Δ_n = n - 1 - offsetDigits n (n! * P(n) - 1).
def shuniaDeficitExact (n : ℕ) : ℕ :=
n - 1 - offsetDigits n (shuniaProd n - 1)
Exact recursive quartering clock step quarterClockStep z = quarterClock z.
def quarterClockStep : ℕ → ℕ
| 0 => 0
| z + 1 => 1 + quarterClockStep (z / 4)
Exact machine counterexample predicate 2 ≤ n ∧ (q_n + 1) * 2 ^ q_n ≤ Δ_n.
def ShuniaCounterexampleExact (n : ℕ) : Prop :=
2 ≤ n ∧
(quarterClockStep (n - 1) + 1) * 2 ^ quarterClockStep (n - 1) ≤
shuniaDeficitExact ntheorem quarterClockStep_eq_quarterClock : ∀ z : ℕ, quarterClockStep z = quarterClock z
⊢ quarterClockStep 0 = quarterClock 0 ⊢ quarterClockStep 0 = quarterClock 0 All goals completed! 🐙
z:ℕ⊢ quarterClockStep (z + 1) = quarterClock (z + 1) z:ℕ⊢ quarterClockStep (z + 1) = quarterClock (z + 1)
z:ℕ⊢ 1 + quarterClockStep (z / 4) = 1 + quarterClock (z / 4)
All goals completed! 🐙theorem shuniaCounterexampleExact_iff {n : ℕ} :
ShuniaCounterexampleExact n ↔ 2 ≤ n ∧ ShuniaCounterexample n := n:ℕ⊢ ShuniaCounterexampleExact n ↔ 2 ≤ n ∧ ShuniaCounterexample n
All goals completed! 🐙theorem shuniaDeficitExact_mul_log_le {n : ℕ} (hn : 2 ≤ n) (hd : 1 ≤ shuniaDeficitExact n) :
(shuniaDeficitExact n : ℝ) * Real.log n ≤ (n : ℝ) - Chebyshev.theta n - 1 :=
shuniaDeficit_mul_log_le hn hdtheorem not_shuniaCounterexampleExact_of_le_21845 {n : ℕ} (hn : n ≤ 21845) :
¬ ShuniaCounterexampleExact n := n:ℕhn:n ≤ 21845⊢ ¬ShuniaCounterexampleExact n
n:ℕhn:n ≤ 21845⊢ ¬(2 ≤ n ∧ ShuniaCounterexample n)
All goals completed! 🐙theorem exists_shuniaCounterexampleExact_iff_not_riemannHypothesis_of_large
(hLarge : RiemannHypothesis → ∀ n : ℕ, 21845 < n →
(n : ℝ) - Chebyshev.theta n ≤
((quarterClockStep (n - 1) + 1) * 2 ^ quarterClockStep (n - 1) : ℝ) * Real.log n) :
(∃ n : ℕ, ShuniaCounterexampleExact n) ↔ ¬ RiemannHypothesis := hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑n⊢ (∃ n, ShuniaCounterexampleExact n) ↔ ¬RiemannHypothesis
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑n⊢ (∃ n, ShuniaCounterexampleExact n) → ¬RiemannHypothesishLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑n⊢ ¬RiemannHypothesis → ∃ n, ShuniaCounterexampleExact n
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑n⊢ (∃ n, ShuniaCounterexampleExact n) → ¬RiemannHypothesis hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑nn:ℕhce:ShuniaCounterexampleExact nhRH:RiemannHypothesis⊢ False
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑nn:ℕhce:ShuniaCounterexampleExact nhRH:RiemannHypothesishsmall:n ≤ 21845⊢ FalsehLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑nn:ℕhce:ShuniaCounterexampleExact nhRH:RiemannHypothesishlarge:21845 < n⊢ False
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑nn:ℕhce:ShuniaCounterexampleExact nhRH:RiemannHypothesishsmall:n ≤ 21845⊢ False All goals completed! 🐙
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑nn:ℕhce:ShuniaCounterexampleExact nhRH:RiemannHypothesishlarge:21845 < n⊢ False hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑nn:ℕhRH:RiemannHypothesishlarge:21845 < nhn2:2 ≤ nhdef:(quarterClockStep (n - 1) + 1) * 2 ^ quarterClockStep (n - 1) ≤ shuniaDeficitExact n⊢ False
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑nn:ℕhRH:RiemannHypothesishlarge:21845 < nhn2:2 ≤ nhdef:(quarterClockStep (n - 1) + 1) * 2 ^ quarterClockStep (n - 1) ≤ shuniaDeficitExact nhpos:1 ≤ shuniaDeficitExact n⊢ False
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑nn:ℕhRH:RiemannHypothesishlarge:21845 < nhn2:2 ≤ nhdef:(quarterClockStep (n - 1) + 1) * 2 ^ quarterClockStep (n - 1) ≤ shuniaDeficitExact nhpos:1 ≤ shuniaDeficitExact nhlog_pos:0 < Real.log ↑n⊢ False
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑nn:ℕhRH:RiemannHypothesishlarge:21845 < nhn2:2 ≤ nhdef:(quarterClockStep (n - 1) + 1) * 2 ^ quarterClockStep (n - 1) ≤ shuniaDeficitExact nhpos:1 ≤ shuniaDeficitExact nhlog_pos:0 < Real.log ↑nhmul:(↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑n ≤ ↑(shuniaDeficitExact n) * Real.log ↑n⊢ False
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑nn:ℕhRH:RiemannHypothesishlarge:21845 < nhn2:2 ≤ nhdef:(quarterClockStep (n - 1) + 1) * 2 ^ quarterClockStep (n - 1) ≤ shuniaDeficitExact nhpos:1 ≤ shuniaDeficitExact nhlog_pos:0 < Real.log ↑nhmul:(↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑n ≤ ↑(shuniaDeficitExact n) * Real.log ↑nh_upper:↑(shuniaDeficitExact n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1⊢ False
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑nn:ℕhRH:RiemannHypothesishlarge:21845 < nhn2:2 ≤ nhdef:(quarterClockStep (n - 1) + 1) * 2 ^ quarterClockStep (n - 1) ≤ shuniaDeficitExact nhpos:1 ≤ shuniaDeficitExact nhlog_pos:0 < Real.log ↑nhmul:(↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑n ≤ ↑(shuniaDeficitExact n) * Real.log ↑nh_upper:↑(shuniaDeficitExact n) * Real.log ↑n ≤ ↑n - Chebyshev.theta ↑n - 1h_rh:↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑n⊢ False
All goals completed! 🐙
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑n⊢ ¬RiemannHypothesis → ∃ n, ShuniaCounterexampleExact n hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑nhNotRH:¬RiemannHypothesis⊢ ∃ n, ShuniaCounterexampleExact n
hLarge:RiemannHypothesis →
∀ (n : ℕ),
21845 < n → ↑n - Chebyshev.theta ↑n ≤ (↑(quarterClockStep (n - 1)) + 1) * 2 ^ quarterClockStep (n - 1) * Real.log ↑nhNotRH:¬RiemannHypothesisn:ℕhn2:n ≥ 2hce:ShuniaCounterexample n⊢ ∃ n, ShuniaCounterexampleExact n
All goals completed! 🐙end Riemann120Analyticend