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module
public import FormalConjecturesUtilErdős Problem 951
References:
[Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
@[expose] public sectionopen scoped Finsupp Nat.Prime Topologyopen Filternamespace Erdos951
A sequence a : ℕ → ℝ is said to have property Erdos951Prop if for any pair of distinct
finitely supported sequences k l : ℕ →₀ ℕ their corresponding Beurling integers are of distance
at least one apart.
def Erdos951Prop (a : ℕ → ℝ) : Prop :=
∀ (k ℓ : ℕ →₀ ℕ), k ≠ ℓ → |beurlingInteger a k - beurlingInteger a ℓ| ≥ 1
If a strictly increasing sequence a : ℕ → ℝ has property Erdos951Prop
and 1 < a 0, then a is a sequence of Beurling prime numbers.
a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)h_diff:a (N + 1) - a N ≥ 1⊢ False
linarith [abs_lt.1 (hN N le_rfl), abs_lt.1 (hN (N + 1) (a:ℕ → ℝha:1 < a 0hm:StrictMono ahe:Erdos951Prop ax:ℝh_contra:¬∃ i, ∀ (a_1 : ℕ), i ≤ a_1 → x ≤ a a_1L:ℝhL:Tendsto a atTop (𝓝 L)N:ℕhN:∀ n ≥ N, dist (a n) L < 1 / 2this:a N < a (N + 1)h_diff:a (N + 1) - a N ≥ 1⊢ N + 1 ≥ N All goals completed! 🐙))]
If 1 < a 0 < ... has property Erdos951Prop, is it true that #{a i ≤ x} ≤ π x?
@[category research open, AMS 11]
theorem erdos_951 : answer(sorry) ↔
∀ a : ℕ → ℝ, 1 < a 0 → StrictMono a → Erdos951Prop a →
∀ᶠ (x : ℝ) in Filter.atTop, {i : ℕ | a i ≤ x}.ncard ≤ π ⌊x⌋₊ := ⊢ True ↔ ∀ (a : ℕ → ℝ), 1 < a 0 → StrictMono a → Erdos951Prop a → ∀ᶠ (x : ℝ) in atTop, {i | a i ≤ x}.ncard ≤ π ⌊x⌋₊
All goals completed! 🐙
Beurling conjectured that if 1 < a 0 < a 1 < ⋯ has property Erdos951Prop and the number
of reals in $[1, x]$ of the form $\prod_i a_i^{k_i}$ is $x + o(\log x)$, then a must be the
sequence of primes. Property Erdos951Prop makes distinct exponent vectors give distinct
Beurling integers, so counting the set BeurlingIntegers a counts the generalised integers.
@[category research solved, AMS 11]
theorem erdos_951.variants.beurling :
∀ a : ℕ → ℝ, IsBeurlingPrimes a → Erdos951Prop a →
((fun x => (BeurlingIntegers a ∩ .Iic x).ncard - x) =o[atTop] Real.log) →
a = Nat.cast ∘ Nat.nth Nat.Prime := ⊢ ∀ (a : ℕ → ℝ),
IsBeurlingPrimes a →
Erdos951Prop a →
(fun x ↦ ↑(BeurlingIntegers a ∩ Set.Iic x).ncard - x) =o[atTop] Real.log → a = Nat.cast ∘ Nat.nth Nat.Prime
All goals completed! 🐙end Erdos951