The number of Hecke eigenvalues of same signs

TL;DR

This paper establishes optimal lower bounds on the count of positive and negative Hecke eigenvalues, enhancing previous results, and also investigates the distribution of these eigenvalues within short intervals.

Contribution

It provides the best possible magnitude bounds for positive and negative Hecke eigenvalues and extends the analysis to short interval cases.

Findings

Established optimal lower bounds for eigenvalues of both signs.

Improved upon previous bounds by Kohnen, Lau & Shparlinski.

Analyzed eigenvalue distribution in short intervals.

Abstract

We give the best possible lower bounds in order of magnitude for the number of positive and negative Hecke eigenvalues. This improves upon a recent work of Kohnen, Lau & Shparlinski. Also, we study an analogous problem for short intervals.