Frontiers of Number Theory

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Questions about properties of positive integers, especially the way in which integers factor and the distribution of prime numbers, have fascinated people for thousands of years and have recently found applications in computer science, information security, and signal processing. This proposal concerns several projects in the theory of numbers, emphasizing connections with other areas of mathematics such as Probability and Combinatorics. The principal investigator was part of a team that made a recent large breakthrough on the distribution of gaps between prime numbers, and further investigations will be made along these lines, together with additional problems about primes and combinatorial objects arising from our techniques. The principal investigator will also continue work on counting integral solutions of certain types of systems of equations, number theoretic functions, and the Riemann zeta function.This research concerns several projects in the theory of numbers. The first concerns accurately counting the integer solutions of certain special types of systems of Diophantine equations that are multidimensional analogs of Vinogradov's system. The chief goal is to obtain near best possible upper bounds for the number of solutions of such systems, for a large class of such systems. The second major project is to improve lower bounds for large gaps between prime numbers and chains of consecutive gaps between primes, and gain more information about the integers within such gaps. The principal investigator will also investigate related problems about the distribution of primes in arithmetic progressions to large moduli and the efficiency of hypergraph coverings, both important tools in research on prime gaps. Further projects include refining the counting function of distinct values taken by Carmichael's function, and studying the distribution modulo 1 of the zeros of the Riemann zeta function.