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

Abstract

In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function \(\zeta (s)\) is not the solution of any algebraic ordinary differential equations on its region of analyticity. In 2015, Van Gorder (J Number Theory 147:778–788, 2015) considered the question of whether \(\zeta (s)\) satisfies a non-algebraic differential equation and showed that it formally satisfies an infinite order linear differential equation. Recently, Prado and Klinger-Logan (J Number Theory 217:422–442, 2020) extended Van Gorder’s result to show that the Hurwitz zeta function \(\zeta (s,a)\) is also formally satisfies a similar differential equation

$$\begin{aligned} T\left[ \zeta (s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. \end{aligned}$$

But unfortunately in the same paper they proved that the operator T applied to Hurwitz zeta function \(\zeta (s,a)\) does not converge at any point in the complex plane \({\mathbb {C}}\). In this paper, by defining \(T_{p}^{a}\), a p-adic analogue of Van Gorder’s operator T,  we establish an analogue of Prado and Klinger-Logan’s differential equation satisfied by \(\zeta _{p,E}(s,a)\) which is the p-adic analogue of the Hurwitz-type Euler zeta functions

$$\begin{aligned} \zeta _E(s,a)=\sum _{n=0}^\infty \frac{(-1)^n}{(n+a)^s}. \end{aligned}$$

In contrast with the complex case, due to the non-archimedean property, the operator \(T_{p}^{a}\) applied to the p-adic Hurwitz-type Euler zeta function \(\zeta _{p,E}(s,a)\) is convergent p-adically in the area of \(s\in {\mathbb {Z}}_{p}\) with \(s\ne 1\) and \(a\in K\) with \(|a|_{p}>1,\) where K is any finite extension of \({\mathbb {Q}}_{p}\) with ramification index over \({\mathbb {Q}}_{p}\) less than \(p-1.\)