On certain zeta functions associated with Beatty sequences

Abstract

Let $\alpha>1$ be an irrational number of finite type $\tau$. In this paper, we introduce and study a zeta function $Z_\alpha^\sharp(r,q;s)$ that is closely related to the Lipschitz-Lerch zeta function and is naturally associated with the Beatty sequence ${\mathcal B}(\alpha):=(\lfloor\alpha m\rfloor)_{m\in{\mathbb N}}$. If $r$ is an element of the lattice ${\mathbb Z}+{\mathbb Z}\alpha^{-1}$, then $Z_\alpha^\sharp(r,q;s)$ continues analytically to the half-plane $\{\sigma>-1/\tau\}$ with its only singularity being a simple pole at $s=1$. If $r\not\in{\mathbb Z}+{\mathbb Z}\alpha^{-1}$, then $Z_\alpha^\sharp(r,q;s)$ extends analytically to the half-plane $\{\sigma>1-1/(2\tau^2)\}$ and has no singularity in that region.