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Analytic continuation of Dirichlet series with almost periodic coefficients. (English) Zbl 1291.30018

Summary: We consider Dirichlet series \(\zeta_{g,\alpha}(s)=\sum_{n=1}^\infty g(n\alpha) e^{-\lambda_n s}\) for fixed irrational \(\alpha\) and periodic functions \(g\). We demonstrate that for Diophantine \(\alpha\) and smooth \(g\), the line \(\text{Re}(s)=0\) is a natural boundary in the Taylor series case \(\lambda_n=n\), 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 \(\sigma_0 = 0\) if \(g\) is odd and real analytic and \(\alpha\) is Diophantine. We show that if \(g\) is odd and has bounded variation and \(\alpha\) is of bounded Diophantine type \(r\), the abscissa of convergence \(\sigma_0\) satisfies \(\sigma_0\leq 1-1/r\). Using a polylogarithm expansion, we prove that if \(g\) is odd and real analytic and \(\alpha\) is Diophantine, then the Dirichlet series \(\zeta_{g,\alpha}(s)\) has an analytic continuation to the entire complex plane.

MSC:

30B40 Analytic continuation of functions of one complex variable
11M41 Other Dirichlet series and zeta functions
30B50 Dirichlet series, exponential series and other series in one complex variable
33E20 Other functions defined by series and integrals
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