Is the completely-multiplicative random partial sum a.s. unbounded relative to N^{1/2}? (Erdős #1144)
Statement
Let $f$ be a random completely multiplicative function: for each prime $p$ choose $f(p)\in\{-1,1\}$ independently and uniformly at random, and extend completely multiplicatively, so $f(mn)=f(m)f(n)$ for all $m,n$ (with no squarefree restriction, in contrast to the Rademacher model). Is it true that $$\limsup_{N\to\infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N}}=\infty$$ with probability $1$?
Acceptance. FULLY RESOLVES: a complete proof (Lean/Coq preferred, else a full written proof) either that the limsup is almost surely $+\infty$ — an almost-sure lower bound showing $\sum_{m\leq N}f(m)/\sqrt{N}$ is unbounded along some subsequence — or that it is almost surely finite (the partial sums are a.s. $O(\sqrt{N})$). ADVANCES: with proof, improve Atherfold's almost-sure upper bound below the $N^{1/2}(\log N)^{1+o(1)}$ stated in background; or prove an almost-sure lower bound of the form $\sum_{m\leq N}f(m)\gg \sqrt{N}\,g(N)$ with $g(N)\to\infty$ along a subsequence. A Monte-Carlo estimate of the normalised limsup is admissible only as clearly flagged evidence, never as a resolution. Anchor any bound claim to the exponent stated in background. Deliver the proof, or the improved bound with proof.
Background
A question of Vaughan [Va99, 1.11]. This completely multiplicative sign model is sometimes itself called a Rademacher random multiplicative function, though that name is often reserved for the merely-multiplicative model supported on squarefree integers, whose partial sums are the subject of Erdős #520 (erdosproblems.com/520); it should also be compared with the Steinhaus model, in which $f(p)$ is uniform on the unit circle. The best known almost-sure upper bound is due to Atherfold [At25], who proved $\sum_{m\leq N}f(m)\ll N^{1/2}(\log N)^{1+o(1)}$ almost surely; whether the normalised limsup is genuinely infinite — equivalently, whether the partial sums are almost surely unbounded when divided by $\sqrt{N}$ — is open. Listed as open on erdosproblems.com/1144 (fetched 2026-07-21, status 'open'); no cash prize; no Lean formalisation yet. The attacker's tool: Monte Carlo simulation of $f$ — sample prime signs, accumulate $\sum_{m\leq N}f(m)$ to large $N$ over many realisations, and estimate the growth of $\max_{N}\lvert\sum_{m\leq N}f(m)\rvert/\sqrt{N}$ — together with analytic work tightening the $\log$-power upper bound or establishing an almost-sure lower bound that grows past $\sqrt{N}$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #1144 (T. F. Bloom) | website |
| REF-02 | Erdős Problem #520 (T. F. Bloom) — Rademacher (squarefree) model | website |
Investigations · 0
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