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

TL;DR

This paper establishes an infinite order linear differential equation for the $p$-adic Hurwitz-type Euler zeta functions, extending previous complex-analytic results to the $p$-adic setting with convergence properties due to non-archimedean norms.

Contribution

It introduces a $p$-adic analogue of Van Gorder's operator and proves a corresponding differential equation for $p$-adic Hurwitz-type Euler zeta functions, demonstrating convergence in the $p$-adic context.

Findings

The $p$-adic operator $T_{p}^{a}$ converges $p$-adically for the Euler zeta functions.

An analogue of Prado and Klinger-Logan's differential equation is established in the $p$-adic setting.

The convergence region is characterized by $s eq 1$ and $|a|_p > 1$ in a finite extension of $Q_p$.

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 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 extended Van Gorder's result to show that the Hurwitz zeta function $\zeta(s,a)$ is also formally satisfies a similar differential equation \begin{equation*}\label{HurDE} T\left[\zeta (s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. \end{equation*} 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{equation*}\label{HEZ} \zeta_E(s,a)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+a)^s}. \end{equation*} 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 $s\neq 1$ 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 $p-1.$