Multiple Zeta Values in Positive Characteristic: A Kaneko-Zagier-Type Conjecture

Abstract

In this paper, we study a comparison between $\infty$-adic and finite multiple zeta values in positive characteristic. We construct a surjective comparison map from the algebra of $\infty$-adic multiple zeta values to a quotient of the algebra of finite multiple zeta values. This map factors through the quotient by ${\zeta }_A(q-1)$, thereby establishing the well-definedness of a Kaneko-Zagier-type conjectural map. We further show that the comparison map is compatible with special values of Carlitz multiple polylogarithms, as well as with the corresponding $\star$-inverse values. As an application, we prove the upper bound predicted by Shi's dimension conjecture for finite multiple zeta values.