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

For a positive irrational number α, we study the ordinary Dirichlet series ζα(s)=∑n≥1⌊αn⌋−s and Sα(s)=∑n≥1(⌈αn⌉−⌈α(n−1)⌉)n−s. We prove relations between them and Jα(s)=∑n≥1({αn}−12)n−s. Motivated by the previous work of Hardy and Littlewood, Hecke and others regarding Jα, we show that ζα and Sα can be continued analytically beyond the imaginary axis except for a simple pole at s=1. Based on the latter results, we also prove that the series ζα(s;β)=∑n≥0(⌊αn⌋+β)−s can be continued analytically beyond the imaginary axis except for a simple pole at s=1.