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Last update: 23/03/2003

 Houches March 2003

École de physique des Houches

B Conrey
L-functions and random matrix theory


Abstract:

Interesting connections have been discovered between the statistics of eigenvalues of groups of random matrices and the statistics of zeros of families of L-functions from number theory. These connections have been pushed further to relate the moments and the value distribution of characteristic polynomials from random matrix theory with the moments and the value distribution of families of L-functions. The purpose of these lectures will be to illustrate these phenomena. Some of the topics covered will be:
  • the basic properties of the Riemann zeta-function
  • an introduction to some families of L-functions
  • some statistics from random matrix theory
  • Katz - Sarnak work on zeros of zeta functions over finite fields
  • theory and numerics about the distributions of zeros of L-functions
  • moments of characteristic polynomials
  • moments of L-functions

lecture: