Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients
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Author(s)
Knill, Oliver
Lesieutre, John D
Date Issued
April 2010
Journal
Complex Analysis and Operator Theory
Publisher
SP Birkhäuser Verlag Basel
Citation
Knill, Oliver, and John Lesieutre. “Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients.” Complex Anal. Oper. Theory 6, no. 1 (April 9, 2010): 237–255.
Version
Author's final manuscript
Abstract
We consider Dirichlet series ζ[subscript g,α](s)=∑[∞ over n=1]g(nα)e[superscript −λ[subscript 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[subscript n] = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series ∑[∞ over n=1]g(nα)z[superscript n] . We prove that a Dirichlet series ζ[subscript g,α](s)=∑[∞ over n=1](nα)/n[superscript s] has an abscissa of convergence σ[subscript 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[subscript 0] satisfies σ[subscript 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[subscript g,α](s) has an analytic continuation to the entire complex plane.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s11785-010-0064-7
