On $\infty$-adic and $v$-adic multiple zeta functions in positive characteristic

TL;DR

This paper develops positive characteristic analogues of $p$-adic multiple zeta functions, providing integral expressions, congruences, and relationships between $ $-adic and $v$-adic multiple zeta functions, advancing understanding in this area.

Contribution

It introduces new integral formulas, congruences, and relationships for $ $-adic and $v$-adic multiple zeta functions in positive characteristic, extending prior $p$-adic results.

Findings

Integral expressions for special values of $ $-adic multiple zeta functions.

Integral formulas for $v$-adic multiple zeta functions similar to $p$-adic cases.

Kummer-type congruences for $v$-adic multiple zeta functions at integers.

Abstract

This paper pursues positive characteristic analogues of the results of Furusho, Komori, Matsumoto and Tsumura on $p$-adic multiple $L$-functions. We consider $\infty$-adic and $v$-adic multiple zeta functions concerned by Angl\`{e}s, Ngo Dac and Tavares Ribeiro. Our main results in this paper consist of: (1) integral expressions of special values of $\infty$-adic multiple zeta functions, (2) integral expression of $v$-adic multiple zeta functions themselves similar to those of $p$-adic multiple $L$-functions, (3) Kummer-type congruence for the special values of $v$-adic multiple zeta functions at integers (4) relationships between special values of $\infty$-adic and those of $v$-adic multipe zeta functions at negative integers and (5) orthognal properties of multiple zeta functions and multiple zeta star functions.