Mathematical Sciences: Multiplicative and Additive Theory ofPrime Numbers and Related Topics in Analytic Number Theory

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This research concerns prime number theory, which can be divided into two approaches, the multiplicative and the additive. The multiplicative theory is largely concerned with the duality between primes and zeros of the Riemann zeta function. The additive theory is based on exponential sums of primes and typically uses characters and Dirichlet L-functions in order to express additive characters as sums of multiplicative functions. The overlap between these two areas is what is being studied in this research. Some specific topics to be examined are pair correlation of zeros of zeta functions, prime pairs and their connection with Kloostermann sums and Barban-Davenpoort type theorems, small gaps between zeta zeros, the function S(T) and Siegal zeros. This research concerns prime number theory, that is, the distribution of the prime numbers in the integers. Certain complex analytic functions, called zeta functions, contain this information and so they are studied uses tools from analysis. Thus this subject is a nice blending of number theory and analysis. The results are very deep and difficult to come by. Goldston has been remarkably successful in solving problems in this area. This research will certainly push forward our knowledge substantially in this important area on mathematics.