论文标题
$ p $ -adic hurwitz型Euler Zeta函数满足的无限订单线性微分方程
Infinite order linear differential equation satisfied by $p$-adic Hurwitz-type Euler zeta functions
论文作者
论文摘要
1900年，在国际数学家大会上，希尔伯特声称Riemann Zeta函数$ζ$不是对其分析性区域的任何代数普通微分方程的解决方案。在2015年，Van Gorder考虑了一个问题，即$ζ$是否满足非代数微分方程，并表明它正式满足无限顺序线性微分方程。最近，Prado和Klinger-Logan Extended Van Gorder的结果表明Hurwitz Zeta函数$ζ（S，A）$也正式满足类似的微分方程\ Begin \ begin {equation*} \ label {hurde} t \ left [ζ（s，a） - \ frac {1} {a^s} \ right] = \ frac {1} {（s-1）a^{s-1}}。 \ end {equation*}，但不幸的是，在同一篇论文中，他们证明了应用于Hurwitz Zeta函数$ζ（S，A）$在复杂平面$ \ Mathbb {C C} $的任何点上都不会收敛。 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 $ζ_{p,E}(s,a)$ which is the $p$-adic analogue of the Hurwitz-type Euler zeta functions \ begin {qore*} \ label {hez}ζ_e（s，a）= \ sum_ {n = 0}^\ infty \ frac {（ - 1）^n} {（n+a）^s}。与复杂情况相反，由于非架构属性，运算符$ t_ {p}^{a} $适用于$ p $ -P $ -ADIC HURWITZ-TYPE EULER ZETA函数$ p，$ p，e} $ s \ in \ mathbb {z} _ {p} $，带有$ s \ neq 1 $和$ a \ in K $，带有$ | a | _ | _ {p}> 1，$ k $是$ k $是$ \ mathbb {q} _ {q} _ {p} $ lamification and lam can y math $ $ \ $ \ p} p} p}的任何有限范围。
In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function $ζ(s)$ is not the solution of any algebraic ordinary differential equations on its region of analyticity. In 2015, Van Gorder considered the question of whether $ζ(s)$ 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 $ζ(s,a)$ is also formally satisfies a similar differential equation \begin{equation*}\label{HurDE} T\left[ζ(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 $ζ(s,a)$ 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 $ζ_{p,E}(s,a)$ which is the $p$-adic analogue of the Hurwitz-type Euler zeta functions \begin{equation*}\label{HEZ} ζ_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 $ζ_{p,E}(s,a)$ 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 $p-1.$
