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

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

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 \(p-1.\)

1900年，希尔伯特在国际数学家大会上声称黎曼zeta函数\(\zeta(s)\)不是任何代数常微分方程在其解析域上的解。 2015年，Van Gorder (J Number Theory 147:778–788, 2015)考虑了\(\zeta (s)\)是否满足非代数微分方程的问题，并证明它形式上满足无限阶线性微分方程。最近，Prado 和 Klinger-Logan (J Number Theory 217:422–442, 2020) 扩展了 Van Gorder 的结果，表明 Hurwitz zeta 函数\(\zeta (s,a)\)也正式满足类似的微分方程

但不幸的是，在同一篇论文中，他们证明了应用于 Hurwitz zeta 函数\(\zeta (s,a)\)的算子T不会在复平面\({\mathbb {C}}\)中的任何点收敛。在本文中，通过定义Van Gorder 算子T的p -adic 模拟\(T_{p}^{a}\)，我们建立了满足\(\zeta _{ p,E}(s,a)\)是Hurwitz 型欧拉 zeta 函数的p进模拟

与复杂情况相反，由于非阿基米德性质，算子\(T_{p}^{a}\)应​​用于p进 Hurwitz 型欧拉 zeta 函数\(\zeta _{p,E }(s,a)\)在\(s\in {\mathbb {Z}}_{p}\)区域内与\(s\ne 1\)和\(a\in K\)且\(|a|_{p}>1,\)其中K是\({\mathbb {Q}}_{p}\)的任何有限扩展，其分支索引为\({\mathbb { Q}}_{p}\)小于\(p-1.\)