Cyclotomic analogues of finite multiple zeta values

TL;DR

This paper introduces cyclotomic analogues of finite multiple zeta values, establishing algebraic relations among them and providing evidence for conjectures linking different types of multiple zeta values.

Contribution

It defines cyclotomic analogues of FMZVs, explores their relations, and supports conjectures connecting FMZVs and SMZVs through algebraic and analytic methods.

Findings

Finite multiple harmonic q-series specialize to FMZVs and SMZVs.

Linear relations among these series induce relations among FMZVs and SMZVs.

Cyclotomic analogues potentially generate a vector space of the same dimension as the original series.

Abstract

We introduce the notion of finite multiple harmonic q-series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetrized multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we obtain families of linear relations among these series which induce linear relations among FMZVs and SMZVs of the same form. This gives evidence towards a conjecture of Kaneko and Zagier relating FMZVs and SMZVs. Motivated by the above results, we define cyclotomic analogues of FMZVs, which conjecturally generate a vector space of the same dimension as that spanned by the finite multiple harmonic q-series at a primitive root of unity of sufficiently large degree.