Properties and applications of Artin L-series

无穷级数的显式评估是数学研究的一个重要方面，它导致了其他科学领域的根本性进步，如工程和计算机科学。一类无穷级数包括和l函数。这些物体对我们理解质数及其分布至关重要。在Artin l -级数的标题下包含的l -级数族包括著名的黎曼ζ函数和狄利克雷l -级数。这些系列的特殊价值，以及研究它们之间的关系，是本提案的中心主题。在最近的出版物中，我和我的合作者在研究这些特殊值方面取得了重大进展。我们已经能够把数学中看似无关的猜想与这个问题联系起来。在未来的工作中，我们希望扩大对这些特殊价值的超越性的认识和理解。

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.