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

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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.