Lower Bounds on High Moments of Twisted Fourier coefficients of Modular Forms

TL;DR

This paper establishes near-optimal lower bounds for high moments of twisted Fourier coefficients of modular forms over primitive Dirichlet characters, advancing understanding of their distribution under the generalized Riemann Hypothesis.

Contribution

It provides the first sharp lower bounds for high moments of twisted Fourier coefficients of modular forms, assuming GRH.

Findings

Lower bounds are sharp up to a constant under GRH.

Results apply to moments of size up to x=1.

Focus on primitive Dirichlet characters and Fourier coefficients.

Abstract

For any large prime $q$, $x \leq 1$ and any real $k\geq 2$, we prove a lower bound for the following $2k$-th moment \begin{equation*} \sum_{\substack{\chi \in X_q^*}} \Big| \sum_{n\leq x} \chi(n)\lambda(n)\Big|^{2k}, \end{equation*} where $X_q^*$ denotes the set of primitive Dirichlet characters modulo $q$ and $\lambda(n)$ the Fourier coefficients of a fixed modular form. The bound we obtain is sharp up to a constant factor under the generalized Riemann Hypothesis.