The typical size of Hecke eigenvalue sums is $o(\sqrt{x})$

Abstract

Let $\mathcal{H}_k$ denote the set of normalized holomorphic Hecke cusp forms of weight $k$ for the full modular group $ \mathrm{SL}_2(\mathbb Z)$. For each $f\in\mathcal{H}_k$, let $\lambda_f(n)$ denote the corresponding eigenvalue of the normalized Hecke operator $T_n$. We prove nontrivial upper bounds for \[\sum_{f \in \mathcal{H}_k}\omega_f \left|\sum_{n \leq x} \lambda_f(n)\right|^{2q},\] where $k$ is a positive even integer, $1\leqslant x\leqslant k$, $0\leqslant q \leqslant 1$ and $ \omega_f = \frac{\Gamma(k-1)} {(4\pi)^{k-1}\langle f,f\rangle} = \frac{2\pi^2} {(k-1)L(1,\mathrm{Sym}^2 f)}. $ Our estimates match the conjecturally sharp upper bound for $1\leqslant x\leqslant k^{0.99}$. In particular, whenever both $x$ and $k/x$ tend to infinity with $k$, we obtain \[\sum_{f \in \mathcal{H}_k}\omega_f \left|\sum_{n \leq x} \lambda_f(n)\right|=o(\sqrt{x}).\] We also prove upper bounds for low moments of sums of Hecke eigenvalues associated with Hecke--Maass cusp forms for $\mathrm{SL}_2(\mathbb Z)$. We study these sums through the probabilistic model proposed by Cogdell--Michel. We also determine the order of magnitude of low moments of this probabilistic model, which is equivalent to determining the order of magnitude of low moments of Hecke eigenvalue sums in the limit.