Partial sums of random multiplicative functions with supercritical divisor twists

Abstract

Let $f$ be a Steinhaus random multiplicative function, and for $\hat{I}\pm \in \mathbb{R}$, let $d_{\hat{I}}\pm$ denote the $\hat{I}\pm$-divisor function. For $\hat{I}\pm \in (1,2)$ we establish that $$\mathbb{E}\left\{\right|\frac{1}{\sqrt{x}}\sum _{n\le x}{d}_{\hat{I}}\pm (n)f(n){|}^{2q}\}\ll \frac{(\log x{)}^{2q(\hat{I}\pm -1)}}{(\log \log x{)}^{3\hat{I}\pm q/2}(1-\hat{I}\pm q)+1}$$ uniformly for $q\in [0,1/\hat{I}\pm ]$ 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 ($\hat{I}\pm =1$) 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 $\hat{I}\pm =1$ (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 $$\underset{T\to \infty }{\lim }\frac{1}{T}{\int }_T^{2T}|\sum _{n\le x}\frac{d_{\hat{I}}\pm (n)}{n^{1/2+it}}{|}^{2q}\mathrm{d}t\ll \frac{(\log x{)}^{2q(\hat{I}\pm -1)}}{(\log \log x{)}^{3\hat{I}\pm q/2}},$$ for $\hat{I}\pm \in (1,2)$ and small $q>0$. This answers a question of Gerspach.

23 pages