Unit cyclotomic multiple zeta values for ${\mu }_2,{\mu }_3$ and ${\mu }_4$

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 ${\left(\zeta \left(\genfrac{}{}{0ex}{}{1}{\epsilon }\right)\right)}^n,{\left(\zeta \left(\genfrac{}{}{0ex}{}{1}{{\epsilon }^{-1}}\right)\right)}^n$ 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 ${\left(\zeta \left(\genfrac{}{}{0ex}{}{1}{\epsilon }\right)\right)}^n,{\left(\zeta \left(\genfrac{}{}{0ex}{}{1}{{\epsilon }^{-1}}\right)\right)}^n$ in the above expressions of unit cyclotomic multiple zeta values.

27 pages

References in corpus (1)

• Standard Relations of Multiple Polylogarithm Values at Roots of Unity