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宣称，黎曼zeta函数\（\ zeta（s）\）并不是解析范围上任何代数常微分方程的解。在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，并证明它正式满足了无穷阶线性微分方程。最近，普拉多（Prado）和克林格·洛根（Klinger-Logan）（J数字理论217：422–442，2020）扩展了范格order（Van Gorder）的结果，以表明Hurwitz zeta函数\（\ zeta（s，a）\）也正式满足相似的微分方程式

但是不幸的是，在同一篇论文中，他们证明了应用于Hurwitz zeta函数\（\ zeta（s，a）\）的算符T不会在复平面\（{\ mathbb {C}} \\}的任何点处收敛。在本文中，通过定义Van Gorder算子T的p -adic类似物\（T_ {p} ^ {a} \），我们建立了由\（\ zeta _ { p，E}（s，a）\），它是Hurwitz型Euler zeta函数的p -adic类似物

与复杂情况相反，由于具有非architemedean属性，运算符\（T_ {p} ^ {a} \）应用于p -adic Hurwitz型Euler zeta函数\（\ zeta _ {p，E }（S，A）\）是会聚的p -adically中的区域\（S \在{\ mathbb {Z}} _ {p} \）与\（S \ NE 1 \）和\（在\ ķ\）与\（| A | _ {p}> 1，\）其中ķ是任何有限延伸\（{\ mathbb {Q}} _ {p} \）与衍生物指数超过\（{\ mathbb { Q}} _ {p} \）小于\（p-1。\）