Valeurs z\^eta multiples

TL;DR

This survey introduces multiple zeta values, tracing their historical development from Euler to modern Galois theory and motives, highlighting their algebraic structure and significance in number theory.

Contribution

It provides a comprehensive overview of multiple zeta values, emphasizing the role of motives and modern Galois theoretic approaches in understanding their structure.

Findings

Historical progression from Euler to modern theories

Role of motives in structuring multiple zeta values

Connection between multiple zeta values and Galois theory

Abstract

This survey article is the written version of two talks given at the Journ\'ees X-UPS 2019 "P\'eriodes et transcendance" at \'Ecole polytechnique. We give a gentle introduction to the study of multiple zeta values, from Euler's solution to the Basel problem to modern Galois theoretic tools. We emphasize the role of the theory of motives which conjecturally controls the structure of the algebra of multiple zeta values.