The Second Moment of Sums of Hecke Eigenvalues II

TL;DR

This paper analyzes the second moment of sums of Hecke eigenvalues for holomorphic cusp forms, revealing different growth behaviors depending on the range of x relative to the weight k.

Contribution

It extends previous work by computing the second moment of Hecke eigenvalue sums over a new range of x, showing a transition in the size of the second moment.

Findings

Second moment ranges between $x^{1/2-o(1)}$ and $x^{1/2}$ in the specified x-range.

Contrasts with earlier results where the second moment was of size $ hickapprox x$ for smaller x.

Provides insights into the distribution of Hecke eigenvalues for large weight forms.

Abstract

Let $f$ be a holomorphic Hecke cusp form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $(\lambda_f(n))_{n\geq 1}$ denote its sequence of normalised 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 range $k^2/(8\pi^2)\leq x\leq k^{12/5-\epsilon}$, the size of the second moment lies between $x^{1/2-o(1)}$ and $x^{1/2}$. This is in sharp contrast to the regime $x\leq k^{2-o(1)}$, where the second moment was shown in preceding work (part I) to be of size $\asymp x$.