Mathematical Sciences: Gaps Between k-Free Numbers, Finite Differences, and Exponential Sums

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This award supports the research in Analytic Number Theory of Professor Michael Filaseta of the University of South Carolina. His project is concerned with gap problems for k-free numbers and in particular for squarefree numbers. His main tool for attacking these problems will be an extension that he has developed to an elementary method of Halberstam and Roth, and to combine this with exponential sum techniques. The field of Analytic Number Theory applies to the discrete realm of the whole numbers the techniques of Analysis, dependent on the notions of continuity and limit, originating in Calculus. The idea of using continuous methods to investigate the discrete is two centuries old, but with the work of the modern analytic number theorists such as Professor Filaseta, the field has had a new rebirth.