An effective criterion for Eulerian multizeta values in positive characteristic

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 us whether any given multizeta value is Eulerian or not. We also prove that if ${\zeta }_A(s_1,\dots ,s_r)$ is Eulerian, then ${\zeta }_A(s_2,\dots ,s_r)$ has to be Eulerian. This was conjectured by Lara Rodríguez 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 can also be extended to determine zeta-like multizeta values.