MEAN VALUES OF L-FUNCTIONS

The Riemann zeta-function and L-functions play a central role in analytic number theory and in mathematics in general. For example, the Riemann zeta-function is defined, for some certain range of the variable, to be a sum over all positive integers n. On the other hand, Euler showed that one can express the zeta-function as a product over the prime numbers. This property immediately underlines the beauty of the theory: it gives a connection between natural numbers and the primes. The problem of determining the properties of prime numbers has a long history, from the ancient theorem of Euclid that there are infinitely many primes, to the celebrated eight page paper of Riemann on the zeta-function in the nineteenth century. Since that time, several important problems in analytic number theory have been solved, and Riemann's ideas have been the inspiration behind much of this progress.Investigating the properties of the zeta-function and L-functions in various contexts leads to many other interesting problems, some of which now represent major challenges in modern mathematics. In fact both the Riemann Hypothesis (which asserts that all the non-trivial zeros of the Riemann zeta-function lie on a particular line), and the Birch & Swinerton-Dyer conjecture (which concerns of the analytic properties of L-functions associated to elliptic curves) have been singled out as two of the seven $1m Clay Mathematics Institute Millennium Prize Problems.My research concerns the ''moments'', or the mean values , of the zeta-function and L-functions, which are averages over families of these functions. These relate to many important and very challenging problems in number theory. The moments of the Riemann zeta-function were first studied by Hardy & Littlewood in 1918, and have been a major theme in analytic number theory since then. More recently the analogous questions for various types of L-function have also been investigated, but progress has been very slow. Even generating conjectures for the asymptotics of these moments has been one of the outstanding problems in the subject over the last hundred years. In 2000, Keating & Snaith used results relating to the moments of the characteristic polynomials of random matrices from various classical compact groups to propose a general conjecture for the asymptotics of all the moments of the zeta-function and families of L-functions. This discovery of connections between the statistical properties of characteristic polynomials and the zeta-functions and L-functions has made significant impact on the field, and is now exciting considerable interest in number theory. Number theory has a tradition of drawing on different areas of mathematics for tools. It is exciting that mathematical physics, and random matrix theory in particular, can now play a role. I propose to concentrate my research on various interesting questions on the mean values of L-functions and also study the analytic properties of certain hybrid Euler-Hadamard products of L-functions. With my project, I anticipate that the approaches will give evidence to underpin the connections between number theory and random matrix theory, providing these two important areas of research with new results, problems and techniques. I expect that it would cast significant new light on some major unsolved problems in number theory.