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问题是问素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zeta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库兹涅佐夫的公式联系起来。 本研究计画主要是将Kuznetsov公式加以推广与修正，以研究二次以上及更复杂L-函数的多项式。 该项目预计将开发强大的新分析工具，推动数论的未来研究。指数和可能出现在不属于SL（2，Z）子群的加法数论问题中。 本研究的目的是通过对SL（n，Z）上的Kuznetsov公式的推广和修正来研究高阶群上的L-函数和指数和的模和。 SL（3，Z）上的超Kloosterman和与非平凡K型的SL（3，Z）自守形式的研究之间似乎有直接的联系，通过SL（3）Kuznetsov公式的修改，所以一个直接的目标是研究这样的形式。 研究SL（3）Kuznetsov公式中的指数和和广义Bessel函数，对于理解SL（3，Z）Maass形式中的L-函数有着重要的意义，因此本研究的目的是在SL（n，Z）（n大于3）的水平方向上继续这一研究.

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.