Polylogarithmic zeta functions and their p-adic analogues

Abstract

We consider a broad family of zeta functions which includes the classical zeta functions of Riemann and Hurwitz, the beta and eta functions of Dirichlet, and the Lerch transcendent, as well as the Arakawa–Kaneko zeta functions and the recently introduced alternating Arakawa–Kaneko zeta functions. We construct their [Formula: see text]-adic analogues and indicate the many strong connections between the complex and [Formula: see text]-adic versions. As applications, we focus on the alternating case and show how certain families of alternating odd harmonic number series can be expressed in terms of Riemann zeta and Dirichlet beta values.