The Second Moment of Sums of Hecke Eigenvalues I

TL;DR

This paper analyzes the average behavior of sums of Hecke eigenvalues for holomorphic cusp forms, revealing size transitions around sums' length comparable to the form's weight and its square.

Contribution

It provides the first and second moment calculations of these sums in the regime where sum length is less than the square of the weight, highlighting size transitions.

Findings

Identifies size transitions at sum lengths around the weight and its square.

Shows the second moment behavior of sums of Hecke eigenvalues.

Prepares ground for future work on larger sum lengths.

Abstract

Let $f$ be a holomorphic Hecke cusp form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $(\lambda_f(n))_{n\geq1}$ denote its sequence of Hecke eigenvalues. We compute the first and second moments of the sums $S(x,f)=\sum_{x\leq n\leq 2x}\lambda_f(n)$, on average over forms $f$ of large weight $k$, in the regime where the length of the sums $x$ is smaller than $k^2$. We observe transitions in the size of the sums when $x\approx k$ and $x\approx k^2$. In subsequent work (part II), it will be shown that once $x$ is larger than $k^2$ (where the latter transition occurs), the average size of the sums $S(x,f)$ becomes dramatically smaller.