Analytic continuation of Dirichlet series with almost periodic coefficients

TL;DR

This paper investigates the convergence and analytic continuation properties of Dirichlet series with coefficients derived from almost periodic functions, establishing conditions under which these series converge or extend analytically.

Contribution

It provides new results on the abscissa of convergence and analytic continuation for Dirichlet series with coefficients involving Diophantine conditions and almost periodic functions.

Findings

Abscissa of convergence is zero for certain analytic coefficients.

Convergence abscissa is bounded by 1-1/r for functions with bounded variation.

Dirichlet series can be analytically continued to the entire complex plane under specific conditions.

Abstract

We prove that an ordinary Dirichlet series with coefficients a(n)=g(n b) has an abscissa of convergence 0 if g is an odd 1-periodic, real-analytic function and b is Diophantine. We also show that if g is odd and has bounded variation and b is of bounded Diophantine type r>1, then the abscissa of convergence is smaller or equal than 1-1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and b is Diophantine, then the ordinary Dirichlet series has an analytic continuation to the entire complex plane.