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

TL;DR

This paper investigates the analytic continuation of Beatty Zeta-Functions and related Dirichlet series, establishing their meromorphic properties and relations inspired by classical work on fractional parts and number theory.

Contribution

It proves the meromorphic continuation of certain Beatty Zeta-Functions and Dirichlet series beyond the imaginary axis, extending previous results and clarifying their pole structure.

Findings

Zeta_ ext{alpha} and S_ ext{alpha} can be continued analytically beyond the imaginary axis.

These functions have a simple pole at s=1.

The series ta_ ext{alpha}(s;eta) also admits meromorphic continuation with a pole at s=1.

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 $s=1.$