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