A Mean Value Theorem for general Dirichlet Series

Abstract

In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)={\sum }_{j=1}^{\infty }{a}_j{n}_j^{-s}$ with positive coefficients for which the counting function $A(x)={\sum }_{n_j\le x}{a}_j$ satisfies $A(x)=\ddot{I}x+O(x^2)$ for some $\ddot{I}>0$ and ${\hat{I}}^2<1$. We prove that $\frac{1}{T}{\int }_0^T|f(\ddot{I}ƒ+it){|}^{2}dt\to {\sum }_{j=1}^{\infty }{a}_j^2{n}_j^{-2\ddot{I}ƒ}$ for $\ddot{I}ƒ>\frac{1+{\hat{I}}^2}{2}$ and obtain an upper bound for this moment for ${\hat{I}}^2<\ddot{I}ƒ\le \frac{1+{\hat{I}}^2}{2}$. We provide a number of examples indicating the sharpness of our results.

To appear in Quart. J. Math