An effective criterion for Eulerian multizeta values in positive characteristic

TL;DR

This paper introduces an effective criterion and algorithm to determine when multizeta values in positive characteristic are Eulerian, linking their properties to rational multiples of the Carlitz period and advancing understanding of their structure.

Contribution

It establishes a new effective criterion and algorithm for identifying Eulerian multizeta values in positive characteristic, confirming conjectures and extending applicability to polylogarithms.

Findings

The criterion characterizes Eulerian multizeta values precisely.

The algorithm can determine Eulerian status for any given multizeta value.

Proves that if a multizeta value is Eulerian, then certain related values must also be Eulerian.

Abstract

Characteristic p multizeta values were initially studied by Thakur, who defined them as analogues of classical multiple zeta values of Euler. In the present paper we establish an effective criterion for Eulerian multizeta values, which characterizes when a multizeta value is a rational multiple of a power of the Carlitz period. The resulting "t-motivic" algorithm can tell whether any given multizeta value is Eulerian or not. We also prove that if zeta_A(s_1,...,s_r) is Eulerian, then zeta_A(s_2,...,s_r) has to be Eulerian. When r=2, this was conjectured (and later on conjectured for arbitrary r) by Lara Rodriguez and Thakur for the zeta-like case from numerical data. Our methods apply equally well to values of Carlitz multiple polylogarithms at algebraic points and zeta-like multizeta values.