Spectral Methods, L-functions and Primes

自同构形式的谱方法是解析数论的基本工具，已有三十多年的历史。最近新的可能性已经打开在该地区的metaplectic形式。该建议提供了大量的扩展的经典结果，这一领域，例如大筛尖形式。亚丛形式的谱理论在二次同余根的分布上的应用正在进行中。这里的要点是，所得到的估计在判别式方面具有很大的一致性。 不太直接，但激励的应用程序的频谱估计结合筛分方法的分布素数。特别是这些可能会产生一个非常好的界的第一个总理完全分裂的希尔伯特类领域。 该提案还包括（与B共同）Conrey和K. Soundararajan）研究了L-函数的零点，证明了在临界线上有相当比例的零点。这个非常基本的问题（对著名的黎曼假设的进步）肯定会吸引研究生。PI在相关研究中与学生和博士后的互动是提案活动不可或缺的一部分。事实上，目前的学生豪尔赫Cantillo在罗格斯大学已经开始工作，以改善密度定理的零点自守L-功能。

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.