Abstract: Multiple zeta values are generalizations of the values of the Riemann zeta function at positive integers. They satisfy many interesting relations and the structure of the relations between them has deep relations with the structure of the Galois group of the rationals. The double shuffle relation is a linear relation between multiple zeta values, obtained by rewriting the product of multiple zeta values as a sum of other such values in two different ways.

Recently, Furusho defined p-adic multiple zeta values. In our work (joint with Furusho) we prove the double shuffle relation for these new values.

In this talk I will survey some of the basics of the theory, essentially due to Euler, explain Furusho's work and give some ideas about the new proof.