Weighted sum formula for multiple zeta values

The sum formula is a basic identity of multiple zeta values that expresses a Riemann zeta value as a homogeneous sum of multiple zeta values of a given dimension. This formula was already known to Euler in the dimension two case, conjectured in the early 1990s for higher dimensions and then proved by Granville and Zagier independently. Recently a weighted form of Euler's formula was obtained by Ohno and Zudilin. We generalize it to a weighted sum formula for multiple zeta values of all dimensions.Comment: 18 page

Structure theorems of mixable shuffle algebras and free commutative Rota-Baxter algebras

We study the ring theoretical structures of mixable shuffle algebras and their associated free commutative Rota-Baxter algebras. For this study we utilize the connection of the mixable shuffle algebras with the overlapping shuffle algebra of Hazewinkel, quasi-shuffle algebras of Hoffman and quasi-symmetric functions. This connection allows us to apply methods and results on shuffle products and Lyndon words on ordered sets. As a result, we obtain structure theorems for a large class of mixable shuffle algebras and free commutative Rota-Baxter algebras with various coefficient rings.Comment: 30 pages. Corrected typos and improved presentatio

Hilbert modular forms and class numbers

In 1975, Goldfeld gave an effective solution to Gauss's conjecture on the class numbers of imaginary quadratic fields. In this paper, we generalize Goldfeld's theorem to the setting of totally real number fields.Comment: 35 page

Double shuffle relations and renormalization of multiple zeta values

In this paper we present some of the recent progresses in multiple zeta values (MZVs). We review the double shuffle relations for convergent MZVs and summarize generalizations of the sum formula and the decomposition formula of Euler for MZVs. We then discuss how to apply methods borrowed from renormalization in quantum field theory and from pseudodifferential calculus to partially extend the double shuffle relations to divergent MZVs