Analytic Theory of L-functions

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L-functions are the basic functions of number theory and are fundamental in the study of prime numbers, solutions to equations in whole numbers, and the uniformity of the distribution of arithmetic sequences. The first L-function is the Riemann zeta-function. The location of its zeros is the subject of the Riemann Hypothesis which is widely regarded as the most important unsolved problem in all of mathmatics. In this project the PI will study various approaches to the Riemann Hypothesis. In addition, he will investigate some very specific statistical properties of the Riemann zeta-function and of families of L-functions in order to more fully understand thes mysterious functions.The PI and his collaborators intend to prove that most of the zeros of Dirichlet L-functions are on the critical line. They also will to develop a general tool, the asymptotic large sieve, to address problems involving averages over all primitive Dirichlet characters of modulus less than a given parameter. They will use this to investigate the spacings between zeros of Dirichlet L-functions. Another project is to determine (conjecturally) the arithemetic part of the distribution of spacings between consecutive zeros of the Riemann zeta-function.Finally, the PI would like to prove the reciprocity formula that he conjectured for Vasyunn sums, and to investigate its relevance in the Nyman-Beurling approach to the Riemann Hypothesis.