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 $A(x)=\rho x + O(x^\beta)$ for some $\rho>0$ and $\beta<1$. We prove that $\frac1T\int_0^T |f(\sigma+it)|^2\, dt \to \sum_{j=1}^\infty a_j^2n_j^{-2\sigma}$ for $\sigma>\frac{1+\beta}{2}$ and obtain an upper bound for this moment for $\beta<\sigma\le \frac{1+\beta}{2}$. We provide a number of examples indicating the sharpness of our results.