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 $Z_{\alpha }^{♯}(r,q;s)$ 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 $Z_{\alpha }^{♯}(r,q;s)$ continues analytically to the half-plane $\{\sigma >-1/\tau \}$ with its only singularity being a simple pole at $s=1$. If $r\notin \mathbb{Z}+\mathbb{Z}{\alpha }^{-1}$, then $Z_{\alpha }^{♯}(r,q;s)$ extends analytically to the half-plane $\{\sigma >1-1/(2{\tau }^2)\}$ and has no singularity in that region.

13 pages