Oscillations of coefficients of Dirichlet series attached to automorphic forms

TL;DR

This paper improves bounds on the oscillations of coefficients of automorphic L-functions and provides results on the sign changes of these coefficients, advancing understanding of their distribution and oscillatory behavior.

Contribution

It refines previous bounds on sums of automorphic coefficients and establishes quantitative results on sign changes for certain automorphic L-functions.

Findings

Enhanced bounds for sums of automorphic coefficients.

Quantitative results on sign changes of real automorphic coefficients.

Improved understanding of oscillatory behavior of automorphic L-function coefficients.

Abstract

For $m\ge 2$, let $\pi$ be an irreducible cuspidal automorphic representation of $GL_m(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. Let $a_\pi(n)$ be the $n^{th}$ coefficient of the $L$-function attached to $\pi$. Goldfeld and Sengupta have recently obtained a bound for $\sum_{n\le x} a_\pi(n)$ as $x \rightarrow \infty$. For $m\ge 3$ and $\pi$ not a symmetric power of a $GL_2(\mathbb{A}_{\mathbb{Q}})$-cuspidal automorphic representation with not all finite primes unramified for $\pi$, their bound is better than all previous bounds. In this paper, we further improve the bound of Golfeld and Sengupta. We also prove a quantitative result for the number of sign changes of the coefficients of certain automorphic $L$-functions, provided the coefficients are real numbers.