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

数论中一些最有趣和最古老的悬而未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着黎曼·泽塔函数和类似的函数，称为L函数。这两个领域在解析数论中通过N.V.库兹涅佐夫的公式联系在一起。本研究主要是推广和修正库兹涅佐夫公式，以研究两个以上的高次多项式和更复杂的L函数。该项目预计将开发出强大的新分析工具，推动未来数论的研究。指数和可能出现在不依附于SL(2，Z)的子群的加法数论问题中。本研究的目的是通过推广和修正SL(n，Z)上的Kuznetsov公式，研究高秩群上指数和的L函数和模和。通过对SL(3)Kuznetsov公式的修正，SL(3，Z)上的超Klosterman和与研究具有非平凡K型的SL(3，Z)自同构型之间似乎有直接的联系，因此一个直接的目的就是研究这种形式。研究SL(3)Kuznetsov公式中的指数和和与广义Bessel函数的关系，使我们对SL(3，Z)Maas型上的L函数的理解有了很大的进步，因此本研究的目的是在水平方向上继续这方面的研究，并对n大于3的SL(n，Z)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.