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 2012 |
All Science Journal Classification (ASJC) codes
- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics