Does the Rademacher random multiplicative partial sum obey an iterated-logarithm law? (Erdős #520)
Statement
Let $f$ be a Rademacher multiplicative function: for each prime $p$ choose $f(p)\in\{-1,1\}$ independently and uniformly at random, extend to squarefree $n=p_1\cdots p_r$ by $f(n)=f(p_1)\cdots f(p_r)$, and set $f(n)=0$ when $n$ is not squarefree. Does there exist a constant $c>0$ such that, almost surely, $$\limsup_{N\to\infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N\log\log N}}=c?$$
Acceptance. FULLY RESOLVES: a complete proof (Lean/Coq preferred, else a full written proof) either that a constant $c>0$ exists with the stated iterated-logarithm law (identifying $c$), or that no such constant exists — for instance by proving Harper's predicted $N^{1/2}(\log\log N)^{1/4+o(1)}$ upper bound, which contradicts any positive $c$. ADVANCES: with proof, improve the almost-sure upper bound below the $(\log\log N)^{3/4+o(1)}$ exponent of Caich stated in background, or improve Harper's almost-sure lower bound beyond ruling out $O\!\big(N^{1/2}/(\log\log N)^{5/2+o(1)}\big)$; a numerical/Monte-Carlo estimate of the normalised limsup is admissible only as clearly flagged evidence toward Erdős-vs-Harper, never as a resolution. Anchor any bound claim to the exponents stated in background. Deliver the proof, or the improved exponent with proof.
Background
Asked by Erdős [Er61, p.251]. If one drops multiplicativity and simply assigns $f(m)=\pm1$ independently, the classical law of the iterated logarithm gives exactly this behaviour with $c=\sqrt{2}$ — Erdős's conjecture asks whether the multiplicative structure preserves it. The known almost-sure upper bounds have steadily improved: Wintner [Wi44] proved $\sum_{m\leq N}f(m)\ll N^{1/2+o(1)}$; Erdős sharpened the exponent to $N^{1/2}(\log N)^{O(1)}$; Lau, Tenenbaum, and Wu [LTW13] reached $N^{1/2}(\log\log N)^{2+o(1)}$; and Caich [Ca24b] improved this to $N^{1/2}(\log\log N)^{3/4+o(1)}$. On the lower side, Harper [Ha13] showed the sum is almost surely NOT $O\!\big(N^{1/2}/(\log\log N)^{5/2+o(1)}\big)$, and — strikingly — conjectured that Erdős's conjecture is FALSE, predicting instead that almost surely $\sum_{m\leq N}f(m)\ll N^{1/2}(\log\log N)^{1/4+o(1)}$ (which would force the normalised limsup to be $0$, not a positive constant). The completely multiplicative variant is Erdős #1144 (erdosproblems.com/1144). A formalised statement exists in google-deepmind/formal-conjectures (FormalConjectures/ErdosProblems/520.lean). Listed as open on erdosproblems.com/520 (fetched 2026-07-21, status 'open'); no cash prize. The attacker's tool: high-throughput Monte Carlo simulation of $f$ — sample the prime signs, accumulate $\sum_{m\leq N}f(m)$ to large $N$ across many realisations, and estimate the growth of the normalised maximum to adjudicate the Erdős-$\sqrt{N\log\log N}$ law against Harper's $(\log\log N)^{1/4}$ prediction — alongside analytic work tightening the $\log\log$ exponent.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #520 (T. F. Bloom) | website |
| REF-02 | Lean formalisation of Erdős #520 (formal-conjectures) | website |
| REF-03 | Law of the iterated logarithm (Wikipedia) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.