The number of coefficients of automorphic $L$-functions for $GL_m$ of same signs

TL;DR

This paper establishes non-trivial lower bounds on the counts of positive and negative coefficients of automorphic $L$-functions for $GL_m$, specifically when the associated representation is self-contragredient, revealing sign distribution properties.

Contribution

It provides the first non-trivial lower bounds for the number of positive and negative coefficients in automorphic $L$-functions for $GL_m$ with self-contragredient representations.

Findings

Lower bounds for positive coefficients

Lower bounds for negative coefficients

Sign distribution insights for automorphic $L$-functions

Abstract

Let $\pi$ be an irreducible unitary cuspidal representation for $GL_m({\Bbb A}_{\Bbb Q})$, and let $L(s, \pi)$ be the automorphic $L$-function attached to $\pi$, which has a Dirichlet series expression in the half-plane $\mbox{Re} s>1$. When $\pi$ is self-contragredient, all the coefficients in the Dirichlet series expression are real. In this paper we give non-trivial lower bounds for the number of positive and negative coefficients, respectively.