Supported by the European Commission, Sixth Framework Programme - Marie Curie Conferences and Training Courses - MSCF-CT-2003-503674
This workshop will explore recent interest in utilising random matrix theory to probe the Riemann zeta function and other L-functions. Montgomery and Dyson realised in the 1970s that the zeros of the Riemann zeta function show the same statistical behaviour as the eigenvalues of random matrices, and more recently Katz and Sarnak proposed that the same is true of statistical properties of zeros of L-functions averaged over families. Over the last few years, random matrix theory has provided the first viable conjectures for mean values of zeta and L-functions on the critical line where their complex zeros are believed to lie. To find more such problems where the subjects can be combined to great effect, this gathering of number theorists and random matrix theorists will measure what can be done in random matrix theory against the current interesting problems in number theory.
So far, progress in number theory via random matrix theory has followed the pattern that a rigorous random matrix calculation is made, then the structure of this result leads to a conjecture for the equivalent number theoretic quantity. These various predictions that random matrix theory provides, such as those for averages of L-functions over families, have been established unconditionally . These in turn have lead, for example, to subconvex estimates for special values of L-functions from which the resolution of some long standing problems in number theory and mathematical physics have been achieved. Much more is expected to be achieved along such lines and this is one of the themes of this meeting.
Thus the number theoretical conjectures based on work in random matrix theory have great merit in themselves, but another of the focuses of the conference will be the search for a way to make rigorous the jump from random matrix theory to number theory.