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 $$p-1.$$