A Mean Value Theorem for general Dirichlet Series

TL;DR

This paper establishes a mean value theorem for general Dirichlet series with positive coefficients, relating the average of their squared magnitude to a sum involving coefficients, under certain growth conditions.

Contribution

It extends mean value theorems to a broad class of Dirichlet series with controlled coefficient growth, providing new asymptotic and upper bound results.

Findings

Proves convergence of the mean value integral for (s)^2 in specified domain.

Derives an upper bound for the mean value when (s)^2 in a different domain.

Provides examples demonstrating the sharpness of the theoretical results.

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.