On the absolute convergence of automorphic Dirichlet series

TL;DR

This paper establishes a lower bound on the abscissa of absolute convergence for automorphic Dirichlet series, linking it to the degree of the series and extending understanding of their convergence properties.

Contribution

It proves a new lower bound on the abscissa of absolute convergence for a broad class of automorphic Dirichlet series, including those in the extended Selberg class.

Findings

Lower bound on the sum of coefficients: (X^{1/2 + 1/(2d)})

Abscissa of absolute convergence satisfies   1/2 + 1/(2d)

Includes automorphic L-functions on GL_n/K

Abstract

Let $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ be a Dirichlet series in the axiomatically defined class ${\mathfrak A}^{\#}$ . The class ${\mathfrak A}^{\#}$ is known to contain the extended Selberg class ${\mathcal S}^{\#}$, as well as all the $L$-functions of automorphic forms on $GL_n/K$, where $K$ is a number field. Let $d$ be the degree of $F(s)$. We show that $\sum_{n<X}|a_n|=\Omega(X^{\frac{1}{2}+\frac{1}{2d}})$, and hence, that the abscissa of absolute convergence of $\sigma_a$ of $F(s)$ must satisfy $\sigma_a\ge 1/2+1/2d$.