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

这项研究涉及数论中众所周知的zeta函数的性质。Carlitz开发了一个zeta函数的类似物，在与Anderson的合作中，这个项目的研究者已经证明了这个广义zeta函数的值是对数的，因此是超越的。这项研究涉及研究者将这些结果扩展到一般功能领域的努力。总的来说，这项研究探索了数论和几何之间的各种类比。研究高斯和的质因数分解及其在函数域的类似分解。对于一般函数域，因式分解不遵循有理函数域的经典模式。

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.