On sum relations of zeta functions

Abstract

The Riemann zeta function is an important function in mathematics. In particular, the Riemann Hypothesis is a conjecture about the roots of this function. A multiple zeta value (MZV) can be viewed as a generalization of the Riemann zeta function which is defined only for positive integer. The history of MZVs dated back to the time of Euler. Indeed, it was a problem proposed by Goldbach in a letter to Euler in an attempt to evaluate double zeta star values in terms of the Riemann zeta function at positive integer. In recent year, the theory of MZVs has opened up quite a lot connections to other branches of mathematics, such as the knot theory, measure theory and the theory of motives. Also, mathematical physicists need relations among MZVs to express some Feynman integrals. Our principle in the study of this theory is to find all algebraic relations among MZVs. In this talk, I should pay special attention to the sum formula of MZVs and introduce the method of adding additional factors to the integral representation of MZVs, and then show some generalizations of the duality theorem and analogous sum relations.