### 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 language | English (US) |
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Pages (from-to) | 237-255 |

Number of pages | 19 |

Journal | Complex Analysis and Operator Theory |

Volume | 6 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2012 |

### All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics