Unit cyclotomic multiple zeta values for $\mu_2,\mu_3$ and $\mu_4$

TL;DR

This paper demonstrates that unit cyclotomic multiple zeta values for roots of unity of orders 2, 3, and 4 generate all cyclotomic multiple zeta values for these roots, with explicit linear combinations and motivic Galois action analysis.

Contribution

It establishes the generative capacity of unit cyclotomic multiple zeta values for bcN and provides explicit linear combinations in cases N=2,3,4, along with Galois action computations.

Findings

Unit cyclotomic multiple zeta values generate all cyclotomic multiple zeta values for bcN in cases N=2,3,4.

Explicit
d-linear combinations of these values are expressed in terms of
d powers of specific zeta values.

Coefficients in these expressions are computed via motivic Galois action analysis.

Abstract

Denote by $\epsilon$ a primitive root of $N^{th}$-unity. In this paper, we show that the unit cyclotomic multiple zeta values for $\mu_N$ generate all the cyclotomic multiple zeta values for $\mu_N$ in cases $N=2,3,4$. Moreover, the unit cyclotomic multiple zeta values for $\mu_N$ can be written as $\mathbb{Q}$-linear combinations of $\left(\zeta\binom{1}{\epsilon}\right)^n, \left(\zeta\binom{1}{\epsilon^{-1}}\right)^n$ and lower depth terms in each weight $n$ in case of $N=2,3$ and $4$. By detailed analysis of the motivic Galois action, we compute the coefficients of $\left(\zeta\binom{1}{\epsilon}\right)^n, \left(\zeta\binom{1}{\epsilon^{-1}}\right)^n$ in the above expressions of unit cyclotomic multiple zeta values.