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 (r, q; s) that is closely related to the Lipschitz-Lerch zeta function and is naturally associated with the Beatty sequence B(α) . .= (⌊αm⌋) m∈N . If r is an element of the lattice Z + Zα −1 , then Z ♯ α (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 ∈ Z + Zα −1 , then Z ♯ α (r, q; s) extends analytically to the half-plane {σ > 1 − 1/(2τ 2 )} and has no singularity in that region.

On the meromorphic continuation of Beatty Zeta-functions and Sturmian Dirichlet series

“…We conclude with a generalization of our previous results regarding the Beatty Zeta-function, which can be compared with the ones in [1].…”

“…Theorem 1. For every irrational α > 0 and q ∈ Z * , the Dirichlet series g αq defined in (1) can be continued analytically to C − 1 τα except for a simple pole at s = 1 with residue {αq}. If τ α > 1, the vertical line − 1 τ α + iR is the natural boundary for g αq .…”