Problem 520

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 $f(p_1\cdots p_r)=f(p_1)\cdots f(p_r)$ (and $f(n)=0$ if $n$ is not squarefree). 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?

Source: erdosproblems.com/520

Number theory
Probability

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An accepted solution exists. The statement is false.

The site labels the problem OPEN. The derived standing is solved with the claim disproved, through the self-contained Lean development submitted by Sigurd Høystad on 2026-08-04 (claim page (Hoystad, 2026)): this corpus built the port of that development in plby/lean-proofs at its commit of 2026-09-15, written for Lean v4.33.0, and audited the port's two compared theorems, that almost surely $\sum_{m\le N}f(m)/\sqrt{N\log\log N}\to0$ and that no constant $c>0$ is almost surely the limit superior in the question. Two further full claims assert the same sharpening of Caich's almost sure bound to $\sum_{m\le N}f(m)\ll\sqrt N(\log\log N)^{1/4+\varepsilon}$ and are pending: a forum claim of 2026-07-29 produced by GPT 5.6-Pro and submitted by Samuel Korsky, with a partial Lean formalization (claim page (Korsky, 2026)), and the arXiv preprint of Durkan and Pearce-Crump of 2026-07-31, which its abstract states settles Harper's conjecture for both the Steinhaus and the Rademacher model (claim page (Durkan Pearce Crump, 2026)).

Ca24b: R. Caich, Almost sure upper bound for random multiplicative functions. arXiv:2304.00943 (2024).
Ha13: Harper, Adam J., Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function. Ann. Appl. Probab. (2013), 584-616.
LTW13: Lau, Yuk-Kam and Tenenbaum, Gérald and Wu, Jie, On mean values of random multiplicative functions. Proc. Amer. Math. Soc. (2013), 409-420.
Wi44: Wintner, Aurel, Random factorizations and Riemann's hypothesis. Duke Math. J. (1944), 267-275.

Problem 520

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