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 $\zeta_\alpha(s) = \sum\limits_{n\geq1} \lfloor\alpha n\rfloor^{-s}$ and $S_\alpha(s) = \sum\limits_{n\geq1} (\left\lceil\alpha n\right\rceil - \left\lceil \alpha (n-1)\right\rceil){n^{-s}}.$ We prove relations between them and $J_{\boldsymbol{\alpha}}(s)=\sum\limits_{n\geq1}\left(\lbrace\alpha n\rbrace-\frac{1}{2}\right)n^{-s}.$ Motivated by the previous work of Hardy and Littlewood, Hecke and others regarding $J_{\boldsymbol{\alpha}},$ 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 $\zeta_{\alpha}(s;\beta)=\sum\limits_{n\geq0}\left(\lfloor\alpha n\rfloor+\beta\right)^{-s}$ can be continued analytically beyond the imaginary axis except for a simple pole at $s=1.$