Finite and symmetric colored multiple zeta values and multiple harmonic q-series at roots of unity

Abstract

The aim of this paper is to develop the Kaneko-Zagier conjecture for higher levels of multiple zeta values. We introduce finite and symmetric multiple zeta values for arbitrary level and prove that they are obtained from an algebraic and analytic operation for a certain multiple harmonic q-series of level N at primitive roots of unity. Relations for finite and symmetric multiple zeta values for level N , such as reversal, harmonic and linear shuffle relations, are also given.