Non-real Poles and Irregularity of Distribution

TL;DR

This paper investigates how non-real poles of weighted Dirichlet series cause oscillatory error terms in distribution, extending classical methods to analyze irregularities even with infinitely many such poles.

Contribution

It generalizes classical techniques to study oscillations caused by non-real poles in weighted Dirichlet series, including cases with infinitely many poles.

Findings

Non-real poles induce oscillatory error terms.

Oscillations occur even with infinitely many poles sharing the same real part.

Method extends classical prime number theorem analysis.

Abstract

We study the general theory of weighted Dirichlet series and associated summatory functions of their coefficients. We show that any non-real pole leads to oscillatory error terms. This applies even if there are infinitely many non-real poles with the same real part. Further, we consider the case when the non-real poles lie near, but not on, a line. The method of proof is a generalization of classical ideas applied to study the oscillatory behavior of the error term in the prime number theorem.