Atherfold 2025 almost sure bounds weighted sums rademacher

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Christopher Atherfold, Almost sure bounds for weighted sums of Rademacher random multiplicative functions. arXiv preprint (2025). arXiv:2501.11076, doi:10.48550/arXiv.2501.11076. The arXiv record (https://arxiv.org/abs/2501.11076, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

The copy read for this card is arXiv:2501.11076v4 (3 February 2026), whose printed title omits "random" ("Almost sure bounds for weighted sums of Rademacher multiplicative functions"). Here f is a Rademacher random multiplicative function: independent signs f(p) = +-1 on the primes, extended multiplicatively to the squarefree integers and equal to 0 on the others. With M_f(x) = sum_{n <= x} f(n)/sqrt(n), Theorem 1 (p. 2) proves that for every eps > 0, almost surely M_f(x) << (log log x)^(3/4+eps). Theorem 2 (p. 3) proves that for every eps > 0, almost surely the sum of f(n)/sqrt(n) over n <= x with largest prime factor P(n) > sqrt(x) is << (log log x)^(1/4+eps); the author conjectures this bound is sharp and states that the bound of Theorem 1 is not. Theorem 3 (p. 3) proves that there exist arbitrarily large x with |M_f(x)| >> (log log x)^(-1/2). The abstract contrasts this with the Steinhaus case, attributing the difference to the size of the Rademacher Euler product, which lets the multiplicative chaos term dominate.

For problem 1144 the paper is a related result, not an answer. The problem asks about a random completely multiplicative f (so f(n) = +-1 for every n) and the unweighted sums normalized by sqrt(N), while the paper treats the squarefree-supported Rademacher model and sums weighted by n^(-1/2); the problem page lists it among its references. Nothing in it proves or refutes that lim sup of the normalized sums is infinite almost surely.

Source: https://arxiv.org/abs/2501.11076.

Bears on. #1144