Almost sure upper bound for random multiplicative functions
Volume 224 / 2026
Acta Arithmetica 224 (2026), 297-325
MSC: Primary 11N37; Secondary 11K99, 60F15
DOI: 10.4064/aa250225-20-2
Published online: 17 September 2026
Abstract
Let $\varepsilon \gt 0$. Let $f$ be a Steinhaus or Rademacher random multiplicative function. We prove that almost surely, as $x \to +\infty $, $$ \sum _{n \leqslant x} f(n) \ll \sqrt{x} (\log _2 x)^{{3}/{4}+ \varepsilon }.$$
