Infinite order linear differential equation satisfied by p-adic Hurwitz-type Euler zeta functions

Abstract

At the international congress of mathematicians in 1900, Hilbert claimed that the Riemann zeta function ζ(s) is not the solution of any algebraic ordinary differential equations on its region of analyticity. Let T be an infinite order linear differential operator introduced by Van Gorder in 2015. Recently, Prado and Klinger-Logan [8] showed that the Hurwitz zeta function ζ(s, a) formally satisfies the following linear differential equation T ζ(s, a) − 1 a s = 1 (s − 1)a s−1 . Then in [5], by defining T a p , a p-adic analogue of Van Gorder's operator T, we constructed the following convergent infinite order linear differential equation satisfied by the p-adic Hurwitz-type Euler zeta function ζ p,E (s, a) T a p ζ p,E (s, a) − a 1−s = 2010 Mathematics Subject Classification. 11R59, 11M35,11B68.