Partial sums of random multiplicative functions with supercritical divisor twists

Abstract

Let $f$ be a Steinhaus random multiplicative function, and for $\alpha\in \mathbb{R}$, let $d_\alpha$ denote the $\alpha$-divisor function. For $\alpha \in (1,2)$ we establish that $$ \mathbb{E}\bigg\{\Big|\frac{1}{\sqrt{x}}\sum_{n\leq x} d_\alpha(n)f(n)\Big|^{2q}\bigg\} \ll \frac{(\log x)^{2q(\alpha-1)}}{(\log\log x)^{3\alpha q/2}(1-\alpha q)+1} $$ uniformly for $q\in [0,1/\alpha]$ and all large $x$. This matches predictions from the theory of supercritical Gaussian multiplicative chaos, and provides an analogue of a seminal result of Harper corresponding to the critical ($\alpha=1$) case. Our approach is based on studying the measure of level sets of an Euler product associated with $f$, and yields a short proof of Harper's upper bound at $\alpha=1$ (implying Helson's conjecture at $q=1/2$). As an additional application, we obtain a conjecturally sharp bound for the pseudomoments of the Riemann zeta function in a certain parameter range, showing that $$ \lim_{T\to\infty}\frac{1}{T}\int_T^{2T} \bigg|\sum_{n\leq x}\frac{d_\alpha(n)}{n^{1/2+it}}\bigg|^{2q} \mathrm{d}t \ll \frac{(\log x)^{2q(\alpha-1)}}{(\log\log x)^{3\alpha q/2}}, $$ for $\alpha\in (1,2)$ and small $q>0$. This answers a question of Gerspach.