Power sums of Hecke's eigenvalues and application

TL;DR

This paper improves estimates on power sums of Hecke eigenvalues using recent advances in symmetric power results, leading to better bounds on the distribution of signs of these eigenvalues.

Contribution

It introduces sharper bounds on power sums of Hecke eigenvalues by leveraging Kim & Shahidi's recent results, enhancing previous estimates.

Findings

Improved bounds on power sums of Hecke eigenvalues.

Enhanced lower bounds for the count of eigenvalues with the same sign.

Application of higher order symmetric power results to classical problems.

Abstract

We sharpen some estimates of Rankin on power sums of Hecke eigenvalues, by using Kim & Shahidi's recent results on higher order symmetric powers. As an application, we improve Kohnen, Lau & Shparlinski's lower bound for the number of Hecke eigenvalues of same signs.