L-Functions, the Kuznetsov Formula, and Exponential Sums in Higher Rank

数论中一些最有趣和最古老的未解决的问题是质数作为多项式的输出出现的频率。另一组有趣的问题围绕黎曼ζ函数和类似的函数，称为l函数。在解析数论中，这两个领域通过库兹涅佐夫的一个公式联系起来。本课题的研究重点是对库兹涅佐夫公式的推广和修正，以研究二次以上复数l函数的多项式。该项目有望开发出强大的新分析工具，推动数论的未来研究。指数和可能出现在不附属于SL（2，Z）子群的可加数论问题中。本研究的目的是通过对SL（n，Z）上的库兹涅佐夫公式的推广和修正来解决高阶群上的l函数和指数和的模和问题。SL（3，Z）上的超kloosterman和与SL（3，Z）非平凡k型自同构形式的研究（通过对SL(3) Kuznetsov公式的修正）之间似乎有直接的联系，因此研究这种形式是一个直接的目标。研究SL(3) Kuznetsov公式中出现的指数和和广义贝塞尔函数，使得对SL（3，Z）质量形式所附l函数的理解取得了重大进展，因此本研究旨在继续在水平方向上以及在n大于3的SL（n，Z）上进行研究。

Some of the most interesting and oldest unanswered problems in number theory ask how often prime numbers appear as the outputs of a polynomial. Another intriguing set of questions surrounds the Riemann zeta function and similar functions, called L-functions. These two areas are connected in analytic number theory through a formula of N.V. Kuznetsov. This research project centers on generalizing and modifying the formula of Kuznetsov to investigate polynomials of degree higher than two and more complex L-functions. The project is expected to develop powerful new analytic tools that will advance future research in number theory. Exponential sums may arise in additive number theory problems that are not attached to subgroups of SL(2,Z). The goal of this research is to address L-functions and moduli sums of exponential sums on higher rank groups via generalizations and modifications of the Kuznetsov formulae on SL(n,Z). There appears to be a direct connection between hyper-Kloosterman sums on SL(3,Z) and the study of SL(3,Z) automorphic forms with non-trivial K-types, by a modification of the SL(3) Kuznetsov formula, so one immediate goal is to study such forms. Studying the exponential sums and generalized Bessel functions occuring in the SL(3) Kuznetsov formula has led to significant advances in understanding the L-functions attached to SL(3,Z) Maass forms, so this research aims to continue this study in the level direction and on SL(n,Z) for n higher than three.