On linear relations among totally odd multiple zeta values related to period polynomials

We show that there is a relationship between modular forms and totally odd multiple zeta values, by relating the matrix $E_{N,r}$, whose entries are given by the polynomial representations of the Ihara action, with even period polynomials. We also consider the matrix $C_{N,r}$ defined by Brown and give a new upper bound of the rank of $C_{N,4}$. This result gives support to the uneven part of the motivic Broadhurst-Kreimer conjecture of depth 4.Comment: 34 pages, to appear in Kyushu J. Mat

Relationships between multiple zeta values of depths 2 and 3 and period polynomials

Some combinatorial aspects of relations between multiple zeta values of depths 2 and 3 and period polynomials are discussed.Comment: I made a major revisio

On Ecalle's and Brown's polar solutions to the double shuffle equations modulo products

Two explicit sets of solutions to the double shuffle equations modulo products were introduced by Ecalle and Brown respectively. We place the two solutions into the same algebraic framework and compare them. We find that they agree up to and including depth four but differ in depth five by an explicit solution to the linearized double shuffle equations with an exotic pole structure.Comment: 22 pages, final version, to appear in Kyushu J. Mat

Cyclotomic analogues of finite multiple zeta values

We introduce the notion of finite multiple harmonic q-series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetrized multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we obtain families of linear relations among these series which induce linear relations among FMZVs and SMZVs of the same form. This gives evidence towards a conjecture of Kaneko and Zagier relating FMZVs and SMZVs. Motivated by the above results, we define cyclotomic analogues of FMZVs, which conjecturally generate a vector space of the same dimension as that spanned by the finite multiple harmonic q-series at a primitive root of unity of sufficiently large degree.Comment: 23 page

Note on totally odd multiple zeta values

A partial answer to a conjecture about the rank of the matrix CN,r introduced by Francis Brown in the study of totally odd multiple zeta values is given