1. FormalConjectures
  2. ErdosProblems
  3. «520»
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Erdős Problem 520

Reference: erdosproblems.com/520

open MeasureTheory ProbabilityTheory Nat Real Filter namespace Erdos520 variable {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, 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 where

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 = 0

Let $f$ be a Rademacher multiplicative function. Does there exist some constant $c > 0$ such that, almost surely, $$ \limsup_{N \to \infty} \frac{\sum_{m \leq N} f(m)}{\sqrt{N \log \log N}} = c? $$

@[category research open, AMS 11 60] theorem declaration uses 'sorry'erdos_520 : answer(sorry) ↔ ∃ 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 := ⊢ True ↔ ∃ c > 0, ∀ (Ω : Type) [inst : MeasureSpace Ω] [IsProbabilityMeasure ℙ] (f : ℕ → Ω → ℝ), IsRademacherMultiplicative f → ∀ᵐ (ω : Ω), limsup (fun N => ∑ m ∈ Finset.Iic N, f m ω / √(↑N * Real.log (Real.log ↑N))) atTop = c All goals completed! 🐙 end Erdos520