Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients

Oliver Knill, John Lesieutre

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

We consider Dirichlet series for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λ n = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series ∑ n=1 g(nα) z n. We prove that a Dirichlet series has an abscissa of convergence σ 0 = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence σ 0 satisfies σ 0 ≤ 1 - 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ g,α(s) has an analytic continuation to the entire complex plane.

Original languageEnglish (US)
Pages (from-to)237-255
Number of pages19
JournalComplex Analysis and Operator Theory
Volume6
Issue number1
DOIs
StatePublished - Feb 1 2012

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics

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