Spectral Methods, L-functions and Primes

自投影型谱方法是三十多年来解析数论的基本工具。最近，在变分形式领域中出现了新的可能性。该提案提出将经典结果实质性地扩展到这一领域，例如用于尖形的大筛子。在二次同余根的分布中，亚可解形式谱理论的应用正在进行中。这里的要点是，所得到的估计相对于判别式是非常一致的。不那么直接，但令人振奋的是将谱估计与筛法结合起来应用于素数的分布。特别是，对于在希尔伯特类域中完全分裂的第一个素数，它们可以产生一个非常好的界。该建议还包括(与B.Conrey和K.Soundarajan共同)研究L函数的零点，证明了此类零点中有相当比例位于临界线上。这个非常基本的问题(迈向著名的黎曼假说的进步)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中不可或缺的一部分。事实上，罗格斯大学的在校生豪尔赫·坎蒂洛已经开始致力于改进自同构L函数的零点的密度定理。

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.