Low moments of random multiplicative functions twisted by Fourier coefficients of modular forms

TL;DR

This paper analyzes the expected magnitude of sums involving Fourier coefficients of modular forms twisted by random multiplicative functions, providing order of magnitude results for certain ranges.

Contribution

It determines the order of magnitude of moments of sums of Fourier coefficients twisted by random multiplicative functions for specific parameters.

Findings

Established the order of magnitude of the expected value of the sum's 2q-th power.

Focused on the case where 0 ≤ q ≤ 1.

Analyzed sums involving Fourier coefficients of modular forms and random multiplicative functions.

Abstract

Let $\lambda(n)$ denote the Fourier coefficients of a fixed modular form and $h(n)$ a Steinhaus or Rademacher random multiplicative function. In this paper, we determine the order of magnitude of \[ \E|\sum_{n \leq x} h(n)\lambda(n)|^{2q} \] for real $x$, $q$ with $0 \leq q \leq 1$.