We study the double shuffle relations satisfied by the double zeta values of level 2, and introduce the double Eisenstein series of level 2 which satisfy the double shuffle relations. We connect the double Eisenstein series to modular forms of level 2.

We study the values 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 symmetric multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we prove the duality formula for these values, as an example of linear relations, which induce those among FMZVs and SMZVs simultaneously. 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.

No abstract

Abstract. 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.

We investigate linear relations among a class of iterated integrals on the Riemann sphere minus four points 0, 1, z and ∞. Generalization of the duality formula and the sum formula for multiple zeta values to the iterated integrals are given.