A Dirichlet series related to the error term in the Prime Number Theorem

Abstract

For a natural number [Formula: see text], let [Formula: see text] where the sum runs over the nontrivial zeros of the Riemann zeta function. For a primitive Dirichlet character [Formula: see text] modulo [Formula: see text], we define [Formula: see text] for [Formula: see text] and obtain the meromorphic continuation of the function [Formula: see text] to the region [Formula: see text]. Our main result indicates that the poles of [Formula: see text] in the region [Formula: see text], if they exist, are related to the zeros of many Dirichlet [Formula: see text]-functions in the same region.