A note on the number of coefficients of automorphic $L-$functions for $GL_m$ with same signs

TL;DR

This paper establishes non-trivial lower bounds on the number of positive and negative coefficients of automorphic $L$-functions for $GL_m$, improving previous results and providing insights into their sign distribution.

Contribution

It provides new lower bounds for the counts of positive and negative coefficients of automorphic $L$-functions for $GL_m$, advancing understanding of their sign patterns.

Findings

Derived non-trivial lower bounds for positive coefficients

Derived non-trivial lower bounds for negative coefficients

Improved upon recent results by Liu and Wu

Abstract

Let $\pi$ be an irreducible unitary cuspidal representation of $GL_m({\Bbb A}_{\Bbb Q})$ and $L(s,\,\pi)$ be the global $L-$function attached to $\pi$. If ${\rm Re}(s)>1$, $L(s,\,\pi)$ has a Dirichlet series expression. When $\pi$ is self-contragradient, all the coefficients of Dirichlet series are real. In this note, we shall give non-trivial lower bounds for the number of positive and negative coefficients respectively, which is an improvement on the recent work of Jianya Liu and Jie Wu.