On certain zeta functions associated with Beatty sequences

Abstract

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