On the meromorphic continuation of Beatty Zeta-Functions and Sturmian Dirichlet series

Abstract

For a positive irrational number $\alpha ,$ we study the ordinary Dirichlet series ${\zeta }_{\alpha }(s)=\sum _{n\ge 1}\lfloor \alpha n{\rfloor }^{-s}$ and $S_{\alpha }(s)=\sum _{n\ge 1}(\lceil \alpha n\rceil -\lceil \alpha (n-1)\rceil )n^{-s}.$ We prove relations between them and $J_{\mathbf{\alpha }}(s)=\sum _{n\ge 1}\left(\{\alpha n\}-\frac{1}{2}\right)n^{-s}.$ Motivated by the previous work of Hardy and Littlewood, Hecke and others regarding $J_{\mathbf{\alpha }},$ we show that ${\zeta }_{\alpha }$ and $S_{\alpha }$ can be continued analytically beyond the imaginary axis except for a simple pole at $s=1.$ Based on the latter results, we also prove that the series ${\zeta }_{\alpha }(s;\beta )=\sum _{n\ge 0}{\left(\lfloor \alpha n\rfloor +\beta \right)}^{-s}$ can be continued analytically beyond the imaginary axis except for a simple pole at $s=1.$

17 pages

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