# Large sums of Hecke eigenvalues of holomorphic cusp forms

## TL;DR

This paper studies the behavior of sums of Hecke eigenvalues of holomorphic cusp forms, establishing conditions under which these sums exhibit cancellations or large deviations, with implications for sign changes and automorphic form families.

## Contribution

It provides new bounds and conditions for cancellations in Hecke eigenvalue sums, extending results analogous to character sums to automorphic forms and analyzing the range of $x$ relative to the weight $k$.

## Key findings

- $S_f(x)=o(x ext{log} x)$ implies some eigenvalues are negative.
- Under RH, $S_f(x)=o(x ext{log} x)$ holds for $x$ growing faster than $ ext{log}k/ ext{log} ext{log}k$.
- Existence of many forms with large sums $S_f(x) ext{gg} x ext{log} x$ when $x=( ext{log}k)^A$.

## Abstract

Let $f$ be a Hecke cusp form of weight $k$ for the full modular group, and let $\{\lambda_f(n)\}_{n\geq 1}$ be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of $\lambda_f(n)$, we investigate the range of $x$ (in terms of $k$) for which there are cancellations in the sum $S_f(x)=\sum_{n\leq x} \lambda_f(n)$. We first show that $S_f(x)=o(x\log x)$ implies that $\lambda_f(n)<0$ for some $n\leq x$. We also prove that $S_f(x)=o(x\log x)$ in the range $\log x/\log\log k\to \infty$ assuming the Riemann hypothesis for $L(s, f)$, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms $f$ of large weight $k$, for which $S_f(x)\gg_A x\log x$, when $x=(\log k)^A.$ Our results are $GL_2$ analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.