The investigator and his collaborators propose to study some problems in analytic number theory. Specifically, they seek to prove an asymptotic formula for the large sieve inequality; this would have applications to small gaps between zeros of Dirichlet L-functions and to mean value formulae for sixth and eighth moments of Dirichlet L-functions. Further, they want to show that there are infinitely many Dirichlet L-functions associated with real primitive characters that have no positive real zeros. This assertion is not known to hold for any infinite family of L-functions. Also, they will investigate the Balazard-Saias criterion for the Riemann Hypothesis. For example, is this criterion equivalent to the Riemann Hypothesis?
Number theory is a branch of mathematics that deals with problems involving whole numbers. Applications of number theory to coding, networking, data compression, and cryptography have thrust number theory into the forefront of modern mathematics. L-functions provide an important tool for investigating many questions of number theory, yet our understanding of the fundamental properties of L-functions is incomplete. The investigator and his collaborators seek to fill in some of the gaps in our basic knowledge of L-functions.
