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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2111.11911 |
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| _version_ | 1866914962931187712 |
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| author | Hu, Su Kim, Min-Soo |
| author_facet | Hu, Su Kim, Min-Soo |
| contents | At the international congress of mathematicians in 1900, Hilbert claimed that the Riemann zeta function $ζ(s)$ is not the solution of any algebraic ordinary differential equations on its region of analyticity. Let $T$ be an infinite order linear differential operator introduced by Van Gorder in 2015. Recently, Prado and Klinger-Logan (J. Number Theory 217: 422--442, 2020) showed that the Hurwitz zeta function $ζ(s,a)$ formally satisfies the following linear differential equation $$
T\left[ζ(s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. $$
Then in (Abh. Math. Semin. Univ. Hambg. 91: 117--135, 2021), by defining $T_{p}^{a}$, a $p$-adic analogue of Van Gorder's operator $T,$ we constructed the following convergent infinite order linear differential equation satisfied by the $p$-adic Hurwitz-type Euler zeta function $ζ_{p,E}(s,a)$ $$ T_{p}^{a}\left[ζ_{p,E}(s,a)-\langle a\rangle^{1-s}\right] =\frac{1}{s-1}\left(\langle a-1 \rangle^{1-s}-\langle a\rangle^{1-s}\right). $$
In this paper, we consider this problem in the positive characteristic case. That is, by introducing $ζ_{\infty}(s_{0},s,a,n)$, a Hurwitz type refinement of Goss zeta function, and an infinite order linear difference operator $L$, we establish the following difference equation \begin{equation*} L\left[ζ_{\infty}\left(\frac{1}{T},s,a,0\right)\right]=\sum_{γ\in\mathbb{F}_{q}} \frac{1}{\langle a+γ\rangle^{s}}. \end{equation*} |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_11911 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Infinite order linear difference equation satisfied by a refinement of Goss zeta function Hu, Su Kim, Min-Soo Number Theory 11R59, 11M35, 11B68 At the international congress of mathematicians in 1900, Hilbert claimed that the Riemann zeta function $ζ(s)$ is not the solution of any algebraic ordinary differential equations on its region of analyticity. Let $T$ be an infinite order linear differential operator introduced by Van Gorder in 2015. Recently, Prado and Klinger-Logan (J. Number Theory 217: 422--442, 2020) showed that the Hurwitz zeta function $ζ(s,a)$ formally satisfies the following linear differential equation $$ T\left[ζ(s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. $$ Then in (Abh. Math. Semin. Univ. Hambg. 91: 117--135, 2021), by defining $T_{p}^{a}$, a $p$-adic analogue of Van Gorder's operator $T,$ we constructed the following convergent infinite order linear differential equation satisfied by the $p$-adic Hurwitz-type Euler zeta function $ζ_{p,E}(s,a)$ $$ T_{p}^{a}\left[ζ_{p,E}(s,a)-\langle a\rangle^{1-s}\right] =\frac{1}{s-1}\left(\langle a-1 \rangle^{1-s}-\langle a\rangle^{1-s}\right). $$ In this paper, we consider this problem in the positive characteristic case. That is, by introducing $ζ_{\infty}(s_{0},s,a,n)$, a Hurwitz type refinement of Goss zeta function, and an infinite order linear difference operator $L$, we establish the following difference equation \begin{equation*} L\left[ζ_{\infty}\left(\frac{1}{T},s,a,0\right)\right]=\sum_{γ\in\mathbb{F}_{q}} \frac{1}{\langle a+γ\rangle^{s}}. \end{equation*} |
| title | Infinite order linear difference equation satisfied by a refinement of Goss zeta function |
| topic | Number Theory 11R59, 11M35, 11B68 |
| url | https://arxiv.org/abs/2111.11911 |