Riemann Hypothesis for Goss $t$-adic Zeta Function

TL;DR

This paper proves the Riemann hypothesis for Goss $v$-adic zeta functions at primes of degree one in the polynomial ring over finite fields, advancing understanding of function field zeta functions.

Contribution

It provides a proof of the Riemann hypothesis for Goss $v$-adic zeta functions at degree one primes, a case previously unresolved.

Findings

Confirmed the Riemann hypothesis for Goss $v$-adic zeta functions at degree one primes.

Established new techniques for analyzing $v$-adic zeta functions.

Contributed to the broader understanding of zeta functions in function field arithmetic.

Abstract

In this short note, we give a proof of the Riemann hypothesis for Goss $v$-adic zeta function $\zeta_{v}(s)$, when $v$ is a prime of $\mathbb{F}_{q}[t]$ of degree one.