Zeta Functions

September 18 - 22, 2006, Moscow, Russia

Laboratoire J.-V. Poncelet

General

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Program

Solomon Friedberg

Boston College, USA

Multiple Dirichlet series attached to Weyl groups

In this talk I give an introduction to generalizations of the Riemann zeta function called "Weyl group multiple Dirichlet series". Weyl group multiple Dirichlet series are sums in several variables whose coefficients involve Gauss sums and also reflect the combinatorics of a given root system. The earliest examples came from Mellin transforms of Eisenstein series and have been extensively studied over the last 20 years. These functions and their residues unify and generalize a number of examples which have been previously treated individually, often with applications to number theory. I will give an account of some of the major research to date and the opportunities for the future.