This proposal comprises four major projects. The first one concerns the distribution of zeros of L-functions, a vast area of research motivated by the Riemann Hypothesis and its generalizations. The proposer plans to investigate in a multidimensional setting the distribution of fractional parts of imaginary parts of zeros, and to study connections with pair correlation of zeros and primes in short intervals. The second project aims to provide, in the context of modular forms, analogs of a recently discovered measure in the complex plane associated to certain quotients of values of the Dedekind Eta function. The third project is concerned with the approximation of real numbers by rationals whose denominators are perfect powers. Over the past few decades many authors have applied ad hoc methods to obtain nontrivial results in this type of context. The proposer intends to use an entirely new approach, which offers unexpected connections with very deep arithmetic phenomena. The last project aims to provide connections between a classical trace problem of Siegel, character values of finite groups, and Weil numbers.
This project falls in the area of number theory, one of the oldest subjects of mathematics. The proposed research comprises projects in several major areas in number theory that abound with open problems and which have attracted some of the top researchers in the field. Most of the proposed projects are motivated by, and build on, recent work by the proposer and his collaborators that has already yielded promising results. It is hoped that by deploying a variety of powerful methods and tools from number theory to these problems will lead to insights and results that will open the door for further developments, and have broader implications.
