### Multiple Zeta Functions - 8EC - M1

Prerequisites

The course assumes basic linear algebra (vector space, subspace, quotient space, dimension, linear map, matrix), basic analysis in one variable (sequences, series and their convergence properties; continuous or differentiable functions, power series with ratio of convergence), and fair knowledge of algebra (basics of groups, rings, fields, quotient groups, quotient rings, finite fields) and complex analysis (holomorphic functions, Cauchy integral, residue sum formula). There are many textbooks that cover those topics, and it is assumed that students have successfully gone through the related courses during their undergraduate degree.



Aim of the course

To provide a comprehensive introduction to multiple generalizations of Riemann's zeta function and analytic, algebraic, arithmetic, combinatorial methods used in their study.

We start with analysis of Euler's gamma and Riemann's zeta functions, their analytic and arithmetic properties, and then discuss their multiple generalizations. Then we continue with investigation of the multiple zeta functions at positive integers - so-called multiple zeta values (MZVs), also known as multiple harmonic series. Those occur in connection with multiple integrals defining invariants of knots and links, and Drinfeld's work on quantum groups, in quantum field theory and throughout combinatorics. Our principal topic of study then becomes the algebra of MZVs over the rationals (namely, the subtle but beautiful structure of rational relations between them), and we complement this story with q-analogues of MZVs and their finite (modulo a prime) version.

The successful student will emerge from this course with much enhanced analytic, algebraic and combinatoric skills. Especially, regarding understanding of proof techniques of identities; underlying algebraic and combinatorial structures; computational issues regarding MZVs and related (polylogarithmic) functions; applications.



Lecturer