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

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 $0 \leq q \leq 1$.