Unit cyclotomic multiple zeta values for μ2,μ3 and μ4

Abstract

Denote by ε a primitive Nth-root of unity. In this paper, we show that the unit cyclotomic multiple zeta values for μN generate all the cyclotomic multiple zeta values for μN when N=2,3,4. Moreover, the unit cyclotomic multiple zeta values for μN can be written as Q-linear combinations of (ζ(1ε))n,(ζ(1ε−1))n and lower depth terms in each weight n when N=2,3 and 4. By detailed analysis of the motivic Galois action, we compute the coefficients of (ζ(1ε))n,(ζ(1ε−1))n in the above expressions of unit cyclotomic multiple zeta values.