On the absolute convergence of automorphic Dirichlet series

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 $G{L}_n/K$, 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 ${\sigma }_a\ge 1/2+1/2d$.

9 pages