Some conjectures and results about multizeta values for F_q[t]

TL;DR

This paper explores conjectures about expressing products of Carlitz-Goss zeta values as F_p-linear combinations of Thakur's multizeta values over F_q[t], extending known results and providing evidence for these complex identities.

Contribution

It introduces new conjectures and proves hundreds of instances relating multizeta value products to linear combinations, generalizing previous q=2 cases.

Findings

Hundreds of conjectures proved with evidence

New identities involving multizeta values

Extension of classical sum shuffle identities

Abstract

In this paper, we explain several conjectures about how a product of two Carlitz-Goss zeta values can be expressed as a F_p-linear combination of Thakur's multizeta values, generalizing the q=2 case dealt by D. Thakur in Relations between multizeta values for F_q[t]. In contrast to the classical sum shuffle, \zeta(a)\zeta(b)=\zeta(a+b)+\zeta(a,b)+\zeta(b,a), the identities we get are much more involved. Hundreds of instances of these conjectures have been proved and we describe the proof method and the evidence.