Analytic Number Theory and mean values of L-functions

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Analytic Number Theory and mean values of L-functions

Group Number Theory

The Riemann zeta-function and other L-functions play a central role in analytic number theory and in mathematics in general. For example, the Riemann zeta-function satisfies an Euler product, which underlines a connection between the natural numbers and the prime numbers. 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 mid-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 Riemann zeta-function and L-functions in various contexts leads to many other interesting problems, 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 and Swinnerton-Dyer Conjecture, which concerns some properties of the L-functions associated to elliptic curves, have been included in the seven Millennium Prize Problems.

The aim of the project is to study various questions related to the moments of the Riemann zeta-function and L-functions, which are the mean values over certain families of these functions. These questions have applications to the distribution of zeros of the Riemann zeta-function (partial answers to the Riemann Hypothesis), the order of magnitude of L-functions (partial answers to the Lindelof Hypothesis), order of vanishing of L-functions at the central point (analytic progress towards the Birch and Swinnerton-Dyer Conjecture), and many others. There is a remarkable connection between the subject and Random Matrix Theory, an area of Mathematical Physics used to describe complex quantum systems.

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