Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients

Abstract

We consider Dirichlet series \({\zeta_{g,\alpha}(s)=\sum_{n=1}^\infty g(n\alpha) e^{-\lambda_n s}}\) 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 λn = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series \({\sum_{n=1}^{\infty} g(n\alpha) z^n}\). We prove that a Dirichlet series \({\zeta_{g,\alpha}(s) = \sum_{n=1}^{\infty} g(n \alpha)/n^s}\) has an abscissa of convergence σ0 = 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 σ0 satisfies σ0 ≤ 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 ζg,α(s) has an analytic continuation to the entire complex plane.