Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients

摘要

We consider Dirichlet series ζ_(g,α)(s) = Σ_(n=1)~∞ g(nα)e~(-λ_ns) 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 seriesΣ_(n=1)~∞ g(nα)z~n.We prove that a Dirichlet series ζ_(g,α)(s) = Σ_(n=1)~∞ g(nα)~s has an abscissa of convergence σ0 = 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 σ0 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.