Infinite order linear difference equation satisfied by a refinement of Goss zeta function

TL;DR

This paper extends the study of zeta functions by establishing an infinite order linear difference equation for a Goss zeta function refinement in positive characteristic, generalizing previous differential equations in complex and p-adic cases.

Contribution

It introduces a new infinite order linear difference equation for a Goss zeta function refinement in positive characteristic, expanding the scope of zeta function equations beyond complex and p-adic cases.

Findings

Established a difference equation for a Goss zeta function refinement in positive characteristic.

Connected the difference equation to sums over finite fields.

Generalized previous differential equations for zeta functions.

Abstract

At the international congress of mathematicians in 1900, Hilbert claimed that the Riemann zeta function $\zeta(s)$ 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 $\zeta(s,a)$ formally satisfies the following linear differential equation $$ T\left[\zeta (s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. $$ 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 $\zeta_{p,E}(s,a)$ $$ T_{p}^{a}\left[\zeta_{p,E}(s,a)-\langle a\rangle^{1-s}\right] =\frac{1}{s-1}\left(\langle a-1 \rangle^{1-s}-\langle a\rangle^{1-s}\right). $$ In this paper, we consider this problem in the positive characteristic case. That is, by introducing $\zeta_{\infty}(s_{0},s,a,n)$, 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[\zeta_{\infty}\left(\frac{1}{T},s,a,0\right)\right]=\sum_{\gamma\in\mathbb{F}_{q}} \frac{1}{\langle a+\gamma\rangle^{s}}. \end{equation*}