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module
public import FormalConjecturesUtilErdős Problem 520
References:
[Hø26] Høystad, S. W. R., A self-contained Lean 4 proof that Erdős Problem #520 has a negative answer (2026), https://github.com/saasom/Erdos520/blob/v1.1.0/paper/erdos520_note.pdf
[Ha13] Harper, A. J., Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function. Ann. Appl. Probab. 23 (2013), 584–616.
@[expose] public sectionopen MeasureTheory ProbabilityTheory Nat Real Filternamespace Erdos520variable {Ω : Type*} [MeasureSpace Ω] [IsProbabilityMeasure (ℙ : Measure Ω)]A random function $f$ is Rademacher multiplicative if $f(1) = 1$, for each prime $p$, we independently choose $f(p) \in {-1, 1}$ uniformly at random (so each $f(p)$ is a measurable function of the sample point), for each square-free integer $n = p_1 \cdots p_r$, $f(n) = f(p_1) \cdots f(p_r)$, and for each non-squarefree integer $n$, $f(n) = 0$.
structure IsRademacherMultiplicative (f : ℕ → Ω → ℝ) : Prop wherePrime entries are random variables.
measurable_of_prime p : p.Prime → Measurable (f p)Prime entries are independent.
iIndepFun_primes : iIndepFun (fun p : Primes ↦ f p) ℙ
Primes entries are uniformly distributed on {-1, 1}.
prob_of_prime p : p.Prime → ℙ {ω | f p ω = 1} = 1 / 2 ∧ ℙ {ω | f p ω = -1} = 1 / 2
map_one ω : f 1 ω = 1
map_mul_of_coprime a b ω : a.Coprime b → f (a * b) ω = f a ω * f b ω
map_of_not_squarefree n ω : ¬ Squarefree n → f n ω = 0Let $f$ be a Rademacher multiplicative function. Does there exist some constant $c > 0$ such that, almost surely, $$ \limsup_{N \to \infty} \frac{\sum_{m \leq N} f(m)}{\sqrt{N \log \log N}} = c? $$
The answer is no: Høystad [Hø26] (with GPT-5.6 Pro and Claude) proved, following the
Halász–Lau–Tenenbaum–Wu–Caich martingale approach with Harper's low-moment estimates [Ha13],
that almost surely $\sum_{m \le N} f(m) \ll \sqrt{N} (\log \log N)^{1/4 + \eta}$ for every
$\eta > 0$, so the $\limsup$ is $0$ almost surely. The linked formal
proof works with the concrete model ℕ → Bool with the product of fair coins, whose
squarefree-supported Rademacher function is IsRademacherMultiplicative; this refutes the
statement below.
@[category research solved, AMS 11 60, formal_proof using lean4 at
"https://github.com/plby/lean-proofs/blob/8822f7ddef30fadbd92e1c6ab4ed897af356af5e/src/latest/ErdosProblems/Erdos520.lean#L24"]
theorem erdos_520 :
answer(False) ↔ ∃ c > 0, ∀ (Ω : Type) [MeasureSpace Ω] [IsProbabilityMeasure (ℙ : Measure Ω)]
(f : ℕ → Ω → ℝ), IsRademacherMultiplicative f →
∀ᵐ ω, limsup (fun N ↦ ∑ m ≤ N, f m ω / sqrt (N * log (log N))) atTop = c := ⊢ False ↔
∃ c > 0,
∀ (Ω : Type) [inst : MeasureSpace Ω] [IsProbabilityMeasure ℙ] (f : ℕ → Ω → ℝ),
IsRademacherMultiplicative f →
∀ᵐ (ω : Ω), limsup (fun N ↦ ∑ m ≤ N, f m ω / √(↑N * Real.log (Real.log ↑N))) atTop = c
All goals completed! 🐙end Erdos520