Analytic continuation of Dirichlet series with almost periodic coefficients

2008/11/09 by Oliver Knill, Knill, Oliver, John Lesieutre +1
Mathematics · #11M99 #30D99 #33E20 #Analytic Number Theory Research #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #math.CV #msc:11M99 #msc:30D99 #msc:33E20

paper · pdf · doi:10.48550/arxiv.0811.1362

15 pages

arxiv created 2008/11/09 · openalex publication_date 2008/11/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Non-vanishing of Dirichlet series with periodic coefficients
• Almost periodicity and boundary values of Dirichlet series
• One metric result about analytic continuation of some Dirichlet series
• Construction of an Ordinary Dirichlet Series with Convergence beyond the Bohr Strip
• On the linear independence of the special values of a Dirichlet series with periodic coefficients