The goal of this project is to investigate fundamental questions about the behavior of the Riemann zeta-function and other L-functions. The proposer will study the distribution of zeros of the derivative of the zeta-function near vertical lines to the right of the critical line by new methods. Empirical studies suggest that the distribution function of these zeros has the peculiar feature of possessing two local maxima. It may be that the theoretical and numerical studies proposed here will explain this phenomenon. A second project is to prove the recent conjectures of H. L. Montgomery and the proposer concerning large values of the zeta-function at its critical points. These conjectures arose in their study of the geometry of the level curves of the zeta-function through these points. The proposer will also investigate the distribution of a-points of L-functions on the critical line and the proportion of simple a-points to the right of it. A goal here is to support Selberg's conjecture that there are only finitely many a-points on the critical line by showing that, in any case, at most an infinitesimal proportion are. Other problems are to determine the pair correlation function of the zeros of the real and imaginary parts of the zeta-function and to determine how it changes as one approaches the critical line; to calculate discrete moments of the derivative of the zeta-function using the proposer?s hybrid formula for the zeta-function and random matrix theory modeling; to determine the connection between the Alternative Hypothesis for the zeros of the zeta-function and an ?alternative? twin prime conjecture; and, finally, to improve significantly the proposer's recent work on moments of finite Euler products by extending these results to very long products.
The projects described in the proposal are all concerned with fundamental issues in analytic number theory, namely, the analytic and geometric properties of the Riemann zeta-function and other L-functions in the Selberg class. Progress will advance the development of the theory of the zeta-function and these other L-functions in both traditional and new directions. Some of the proposed methods are new and may well have applications to other areas since most of the problems have connections with harmonic analysis, probability theory, and complex function theory. There are also direct connections with questions in random matrix theory such as the distribution of zeros of the derivative of characteristic polynomials of random matrices.
