### 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 |

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### All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

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*Complex Analysis and Operator Theory*, vol. 6, no. 1, pp. 237-255. https://doi.org/10.1007/s11785-010-0064-7

**Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients.** / Knill, Oliver; Lesieutre, John.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients

AU - Knill, Oliver

AU - Lesieutre, John

PY - 2012/2/1

Y1 - 2012/2/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s11785-010-0064-7

DO - 10.1007/s11785-010-0064-7

M3 - Article

AN - SCOPUS:84856222600

VL - 6

SP - 237

EP - 255

JO - Complex Analysis and Operator Theory

JF - Complex Analysis and Operator Theory

SN - 1661-8254

IS - 1

ER -