Zero Distribution of $v-$adic Multiple Zeta Values over $\mathbb{F}_q(t)$

TL;DR

This paper investigates the zero distribution of $v$-adic multiple zeta values over function fields, establishing that they only vanish at trivial zeros for degree one primes and conjecturing a generalization.

Contribution

It proves that $v$-adic MZVs at negative integers vanish only at trivial zeros for degree one primes and proposes a conjecture for all primes.

Findings

$v$-adic MZVs vanish only at trivial zeros for degree one primes

Conjecture that this zero distribution pattern extends to all primes

Provides insight into the structure of $v$-adic multiple zeta values over function fields

Abstract

This paper aims to study the zero distribution of $v-$adic multiple zeta values over function fields. We show that the interpolated $v-$adic MZVs at negative integers only vanish at what we call the ''trivial zeros'', for degree one prime over rational function fields. And we conjecture that this result can be generalized to all primes.