Applications of the Kuznetsov formula on GL(3)

Area of Science:

• Number Theory
• Automorphic Forms
• Representation Theory

Background:

• The Kuznetsov formula is a powerful tool in analytic number theory, relating automorphic forms to number theoretic objects.
• Generalizations and explicit forms of the Kuznetsov formula are crucial for studying various arithmetic phenomena.

Purpose of the Study:

• To develop an explicit version of the Kuznetsov formula specifically for the group GL(3).
• To investigate the analytic properties of the test functions involved in this formula.
• To demonstrate the utility of the new formula through diverse applications.

Main Methods:

• Derivation of an explicit Kuznetsov formula for GL(3).
• Analysis of the analytic behavior of test functions on both sides of the formula.
• Application of the derived formula to specific problems in number theory.

Main Results:

• An explicit Kuznetsov formula for GL(3) has been successfully developed.
• The analytic behavior of the associated test functions has been thoroughly discussed.
• The formula's applicability was demonstrated through connections to Weyl's law, exceptional eigenvalues, large sieve methods, and L-functions.

Conclusions:

• The explicit Kuznetsov formula for GL(3) provides a valuable new tool for number theoretic research.
• The formula's applications highlight its significance in understanding fundamental problems in analytic number theory.
• Further research can explore extensions and additional applications of this explicit formula.