Mathematical Sciences: Gauss Sums, Zeta and Gamma Functions in Arithmetic of Function Fields

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This research concerns the nature of the well-known zeta function in number theory. Carlitz has developed an analogue of the zeta function and in joint work with Anderson, the investigator on this project has shown the values of this generalized zeta function are logarithmic and hence transcendental. This research concerns the effort by the investigator to extend these results to general function fields. In general, this research explores various analogies between number theory and geometry. The research examines the prime factorizations of Gauss sums and the factorization analogues for function fields. For general function fields, the factorizations do not follow the classical patterns for the field of rational functions.