Let $f$ be a Rademacher multiplicative function: a random ${- 1, 0, 1}$ -valued multiplicative function, where for each prime $p$ we independently choose $f (p) \in {- 1, 1}$ uniformly at random, and for square-free integers $n$ we extend $f (p_{1} \dots p_{r}) = f (p_{1}) \dots f (p_{r})$ (and $f (n) = 0$ if $n$ is not squarefree). Does there exist some constant $c > 0$ such that, almost surely, $N \to \infty lim sup \frac{\sum _{m \leq N} f ( m )}{N lo g lo g N} = c ?$

Worked, still open.

number theory · open · formalized (Lean) · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

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unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

This is **open** as of January 2026. In fact, the question in exactly this form is recorded as **Erdős Problem #520** (from Erdős 1961) and is listed as open in current compilations. ([Erdős Problems][1])

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

formal

AMS 11 60 · open (literature)

theorem erdos_520 :
    answer(sorry) ↔ ∃ c > 0, ∀ (f : ℕ → Ω → ℝ), IsRademacherMultiplicative f →
      ∀ᵐ ω, limsup (fun N ↦ ∑ m ≤ N, f m ω / sqrt (N * log (log N))) atTop = c

formal-conjectures/520.lean ↗

links

law of the iterated logarithm · reference

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open

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formal

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