A Mean Value Theorem for general Dirichlet Series

In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)={\sum }_{j=1}^{\mathrm{\infty }}{a}_j{n}_j^{-s}$ 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 $\frac{1}{T}{\int }_0^T\mathrm{\mid }f(\sigma +it){\mathrm{\mid }}^{2}dt\to {\sum }_{j=1}^{\mathrm{\infty }}{a}_j^2{n}_j^{-2\sigma }$ 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.