L-functions: Symmetry and Zeros

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This project concerns the connection between fundamental problems in number theory, zeros of arithmetic L-functions,and random matrix theory. Recent developments have placed the confluence of these fields at the center of a great deal of important work that combines activity and interest from researchers previously working in completely different areas. The study of zeros of arithmetic Dirichlet series, or L-functions, occupies a central position in number theory and includes within its realm both the Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture. The theory of eigenvalues of random matrices is a subject begun nearly 50 years ago by physicists interested in the statistics of energy levels of excited particles. The connection between these two fields was revealed for the first time in 1974 in the paper ``The pair correlation of the zeros of the zeta-function'' by H. L. Montgomery. This project will involve both number theorists and mathematical physicists to develop a comprehensive new model for families of L-functions in order to gain insights into a number of fundamental open problems about L-functions. For example, by combining random matrix theory and number theory appropriately, the principal investigator and his colleagues intend to (i) find all of the terms in asymptotic formulae for moments of families of L-functions, (ii) determine precisely how large the extreme-most values of L-functions can be, and (iii) analyze the vanishing at the central critical point for families of L-functions.In 1859 Bernhard Riemann made a conjecture about an importantfunction known as the zeta-function. Riemann's hypothesis is still unproven and is widely regarded as the most important unsolved problem inmathematics. It has long been known that the solution ofthis problem would have significant applications to cryptography, coding theory, and the study of prime numbers. Recently, it has also been realized that the zeta-function and the closely related L-functions provide models for the behavior of fundamental particles in physics. The project `L-functions: symmetry and zeros' will bring mathematicians and physicists together to explore this amazing connection more deeply through the use of a statistical tool called `random matrix theory.'