On Primitivity and Vanishing of Dirichlet Series

TL;DR

This paper establishes new criteria for when Dirichlet series associated with rational periodic functions vanish at positive integers, advancing understanding of classical conjectures and character decompositions.

Contribution

It provides a novel necessary and sufficient condition for Dirichlet series vanishing and characterizes functions with multiple zeros, using primitive character decomposition.

Findings

Identifies new families of numbers satisfying Erdős's conjecture.

Characterizes functions with Dirichlet series vanishing at two points.

Introduces a decomposition approach for Dirichlet characters.

Abstract

For a rational valued periodic function, we associate a Dirichlet series and provide a new necessary and sufficient condition for the vanishing of this Dirichlet series specialized at positive integers. This question was initiated by Chowla, and carried out by Okada for a particular infinite sum. Our approach relies on the decomposition of the Dirichlet characters in terms of primitive characters. Using this, we find some new family of natural numbers for which a conjecture of Erd\"os holds. We also characterize rational valued periodic functions for which the associated Dirichlet series vanishes at two different positive integers under some additional conditions.