Spectral Methods, L-functions and Primes

点击翻译按钮获取中文摘要

The spectral methods of automorhpic forms are basic tools of analytic number theory for more than three decades. Recently new possibilities have been opened in the area of metaplectic forms. The Proposal offers to make substantial expansion of the classical results to this territory, such as for example the large sieve for cusp forms. Applications of the spectral theory of metaplectic forms to the distribution of roots of quadratic congruences are in progress. Here the point is that the resulting estimates are valid in great uniformity with respect to the discriminant. Less direct, yet motivating are applications of spectral estimates in conjunction with sieve methods to the distribution of prime numbers. In particular these could yield a very good bound for the first prime which splits completely in the Hilbert class field. The proposal also includes (jointly with B. Conrey and K. Soundararajan) study of zeros of L-functions, proving that a respectful percentage of such zeros lay on the critical line. This very basic problem (a progress towards the celebrated Riemann Hypothesis) will definitely attract graduate students. An interaction of the PI in the related research with students and postdoc is an indispensable part of the activity under the Proposal. In fact a current student Jorge Cantillo at Rutgers already began to work on improving the density theorems for the zeros of automorphic L-functions.