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

## TL;DR

This paper introduces finite and symmetric colored multiple zeta values at roots of unity as limits of truncated multiple harmonic q-series, extending the Kaneko-Zagier conjecture to higher levels and cyclotomic cases.

## Contribution

It generalizes the framework of finite and symmetric multiple zeta values to colored and higher level cases using q-series limits, proposing a higher level Kaneko-Zagier conjecture.

## Key findings

- Defined colored multiple zeta values at roots of unity
- Established algebraic and analytic limits of q-series for these values
- Proposed a higher level analogue of the Kaneko-Zagier conjecture

## Abstract

The Kaneko-Zagier conjecture states that finite and symmetric multiple zeta values satisfy the same relations. In the previous work with H.~Bachmann and Y.~Takeyama, we proved that the finite and symmetric multiple zeta value are obtained as an `algebraic' and `analytic' limit at $q\rightarrow 1$ of a certain truncated multiple harmonic $q$-series, and studied its relations in order to give partial evidence of the Kaneko-Zagier conjecture. In this paper, we start with truncated multiple harmonic $q$-series of level $N$, which is a $q$-analogue of the truncated colored multiple zeta value. We introduce our finite and symmetric colored multiple zeta values as an algebraic and analytic limit of the truncated multiple harmonic $q$-series of level $N$ and discuss a higher level (or a cyclotomic) analogue of the Kaneko-Zagier conjecture.