Grant: Discovery Grants Program - Individual

The study of natural numbers (i.e. number theory) has fascinated mathematicians for centuries. Research in number theory involves the application of methods and knowledge from many branches of mathematics. This proposed research is related to the analytic theory of numbers. Analytic number theory is a branch of number theory that uses the techniques of classical analysis to address problems of an arithmetical nature. One of the main themes in this approach is the study of L-functions. L-functions are analytic functions that encode the arithmetical information in the original problem so that analytic techniques can be used. L-functions play a central role in modern number theory and there has been tremendous interest in them. Many important theorems of number theory are intimately connected with certain special values of various L-functions. For example, the celebrated prime number theorem on the distribution of prime numbers is a result of the study of the special values of a certain L-function. In this research I wish to gain more insight into these important objects and the deep conjectures related to them  by considering some problems related to zeros and special values of L-functions. More specifically, one of the main focuses of this research is to study the zeros of L-functions belonging to a certain axiomatically defined class of L-functions, the Selberg class. This research also aims at different applications of L-functions and related analytic techniques in studying several arithmetic functions. More specifically, I would like to obtain  results on size and divisibility properties of some important arithmetic functions. The outcome of this research will enrich the knowledge in this important area of number theory and will help to pave the way for further studies in this field.