Partial sums of random multiplicative functions with supercritical divisor twists

TL;DR

This paper analyzes the behavior of partial sums of random multiplicative functions with divisor twists, establishing new bounds that align with predictions from Gaussian multiplicative chaos theory and extending Harper's results.

Contribution

It provides a novel approach to bounding moments of divisor-twisted random multiplicative functions, matching supercritical chaos predictions and offering a short proof of Harper's upper bound.

Findings

Established bounds for moments of divisor-twisted sums matching chaos predictions

Provided a short proof of Harper's upper bound at the critical case

Derived sharp bounds for pseudomoments of the Riemann zeta function in certain ranges

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.