Automorphic L-functions and Sums of Automorphic L-functions

9970118本课题将研究数论、表示论和调和分析中的L函数。具体地说，该项目将通过系统地研究由L函数的和组成的两个复变量中的某个狄里克莱级数来研究自同构的L函数。这些自然产生的狄里克莱级数本身不是欧拉乘积，但它们的各个系数是欧拉的。其目的是使用这些级数中的额外变量来提取分析信息，类似于通过近似函数方程可以获得的信息，如果它可以进行到无限长的话。这两个变量Dirichlet级数最早出现在Rankin-Selberg积分的研究中。本项目还将继续研究这些积分。在二十世纪后半叶，整数的算术性质在通信、安全和现代生活的其他令人惊讶的领域找到了新的应用。数学家们长期以来一直在数论的一个数学分支中研究整数。整数的一些最简单的性质仍然包含着深刻的奥秘，最近数学家们开发了强大的新技术来探索这些奥秘。这个项目的主要目的是研究表现出某些复杂对称性的函数。这些函数令人感兴趣，因为它们通常编码有关其他事物的算术信息，例如方程组的解的数目。对这些函数的研究提供了对对称性和编码算法的深入了解。

9970118This project will study L-functions, which arise in number theory, representation theory, and harmonic analysis. Specifically, the project will study automorphic L-functions through the systematic examination of certain Dirichlet series in two complex variables built up out of sums of L-functions. These naturally occurring Dirichlet series are not themselves Euler products, but their individual coefficients are Eulerian. The intent is to use the extra variable in these series to extract analytic information similar to that which one could be obtained by the approximate functional equation were it possible to carry it out to infinite length. These two variable Dirichlet series originally appeared in investigations of Rankin-Selberg integrals. This project will also continue the study these integrals.In the last part of the twentieth century, the arithmetic properties of the integers have found new application in communications, security and other surprising areas of modern life. Mathematicians have long studied the integers in a branch of mathematics called number theory. Some of the simplest properties of the integers still contain deep mysteries, and recently mathematicians have developed powerful new techniques to explore these mysteries. The broad aim of this project is to study functions which exhibit certain complicated symmetries. These functions are of interest because they often encode arithmetic information about other things, for instance the number of solutions to sets of equations. The study of these functions provides insight into the symmetries and into the encoded arithmetic.