Local Behavior of Arithmetical Functions with Applications to Automorphic L-Functions

TL;DR

This paper develops a Voronoi-type series approximation for arithmetical functions linked to automorphic L-functions, enabling analysis of oscillation frequencies and sign changes, with applications to various automorphic forms.

Contribution

It introduces a new approximation method for arithmetical functions associated with automorphic L-functions, improving understanding of their oscillations and sign changes.

Findings

Derived a Voronoi-type series approximation for arithmetical functions.

Proved a new result on the non-existence of certain elements in the Selberg class.

Improved bounds on sign-change problems for automorphic L-function coefficients.

Abstract

We derive a Voronoi-type series approximation for the local weighted mean of an arithmetical function that is associated to Dirichlet series satisfying a functional equation with gamma factors. The series is exploited to study the oscillation frequency with a method of Heath-Brown and Tsang [7]. A by-product is another proof for the well-known result of no element in the Selberg class of degree 0 \textless{} d \textless{} 1. Our major applications include the sign-change problem of the coefficients of automorphic L-functions for GL m , which improves significantly some results of Liu and Wu [14]. The cases of modular forms of half-integral weight and Siegel eigenforms are also considered.