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 λ 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 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.
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Communicated by Daniel Aron Alpay.
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Knill, O., Lesieutre, J. Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients. Complex Anal. Oper. Theory 6, 237–255 (2012). https://doi.org/10.1007/s11785-010-0064-7
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DOI: https://doi.org/10.1007/s11785-010-0064-7