On certain zeta functions associated with Beatty sequences

Let $\alpha >1$ be an irrational number of finite type $\tau$. In this paper, we introduce and study a zeta function $Z_{\alpha }^{\mathrm{♯}}(r,q;s)$ that is closely related to the Lipschitz-Lerch zeta function and is naturally associated with the Beatty sequence ${\mathcal B}(\alpha):=(\lfloor\alpha m\rfloor){m\in{\mathbb N}}$\mathrm{.}If$r$isanelementofthelattice${\mathbb Z}+{\mathbb Z}\alpha^{-1}$,then$Z\alpha^\sharp(r,q;s)$ continues analytically to the half-plane $\{\sigma >-1\mathrm{/}\tau \}$ with its only singularity being a simple pole at $s=1$. If $r\in ̸\mathbb{Z}+\mathbb{Z}{\alpha }^{-1}$, then $Z_{\alpha }^{\mathrm{♯}}(r,q;s)$ extends analytically to the half-plane $\{\sigma >1-1\mathrm{/}(2{\tau }^2)\}$ and has no singularity in that region.