Large sums of Hecke eigenvalues of holomorphic cusp forms

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.