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Claim. Benjamin Durkan and Andrew Pearce-Crump, A sharp almost sure upper bound for partial sums of random multiplicative functions, arXiv:2607.29429 (submitted 2026-07-31; a revised version is dated 2026-10-01), prove that for a Steinhaus or a Rademacher random multiplicative function $f$ and every $\varepsilon>0$, almost surely

$$\Bigl\lvert\sum_{n\le x}f(n)\Bigr\rvert \ll_{\varepsilon,f}\sqrt x\,(\log\log x)^{1/4+\varepsilon}.$$

With Harper's almost sure lower bound this fixes the exponent of the iterated logarithm in both models and settles Harper's conjecture on the large fluctuations of random multiplicative functions, which is how the abstract states the result. For the Rademacher model of Problem 520, take $\varepsilon<1/4$: the bound gives $\sum_{m\le N}f(m)/\sqrt{N\log\log N}\to0$ almost surely, so the limit superior in the problem is $0$ and no constant $c>0$ exists. The answer is no. The abstract does not name the problem; the bearing on it is this one-line consequence, and the paper is identified as the same bound as the forum claims on the site's proof-claims page by Høystad, who describes the three routes as essentially the same modification of Caich's argument (card): a conditional high-moment estimate on each thin block of primes in place of a first-moment bound and a union bound. This account follows the arXiv abstract.

Depends on. Nothing in this wiki.

Standing. Pending: an arXiv preprint with no refereed version found and no acceptance recorded by the site, which labels the problem OPEN and whose proof-claims tab did not list the paper. The two forum claims of the same bound are the Korsky claim, posted two days before this preprint, and the Høystad claim, a self-contained Lean development.