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On certain zeta functions associated with Beatty sequences

Volume 185 / 2018

William D. Banks Acta Arithmetica 185 (2018), 233-247 MSC: Primary 11M35; Secondary 11B83. DOI: 10.4064/aa170528-29-3 Published online: 15 June 2018


Let $\alpha>1$ be an irrational number of finite type $\tau$. 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 $\{\sigma>1-1/(2\tau^3)\}$ and has no singularity in that region.


  • William D. BanksDepartment of Mathematics
    University of Missouri
    Columbia, MO 65211, U.S.A.

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