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 $p-1.$