Properties and applications of Artin L-series

点击翻译按钮获取中文摘要

The explicit evaluation of infinite series is an important aspect of mathematical research that has led to fundamental advances in other areas of science, such as engineering and computer science. One class of infinite series includes zeta and L-functions. These objects have been pivotal in our understanding of prime numbers and their distribution. A family of L-series contained under the heading of Artin L-series includes the celebrated Riemann zeta function and Dirichlet's L-series. The special values of these series, as well as the study of relations among them, is the central theme of this proposal. In recent publications, my collaborators and I have been able to make significant progress in the study of these special values. We have been able to connect seemingly unrelated conjectures in mathematics to this problem. In future work, we expect to enlarge our awareness and understanding of the transcendental nature of these special values.