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Infinite order linear differential equation satisfied by $p$-adic Hurwitz-type Euler zeta functions
- Publication Year :
- 2020
- Publisher :
- arXiv, 2020.
-
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 $p-1.$<br />Comment: 18 pages. Final version. Dedicated to the memory of Prof. David Goss (1952-2017)
- Subjects :
- Mathematics - Number Theory
Mathematics::General Mathematics
General Mathematics
Mathematics::Number Theory
010102 general mathematics
Order (ring theory)
Type (model theory)
01 natural sciences
11M35, 11B68
Riemann zeta function
010101 applied mathematics
Combinatorics
Hurwitz zeta function
symbols.namesake
Number theory
Linear differential equation
Mathematics - Classical Analysis and ODEs
Ordinary differential equation
symbols
Classical Analysis and ODEs (math.CA)
FOS: Mathematics
Number Theory (math.NT)
0101 mathematics
Complex plane
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....89db41846fe00eedd644a71b5b0ff820
- Full Text :
- https://doi.org/10.48550/arxiv.2008.07218