Spectral Theory of the Riemann Zeta-Function: Chapter 6: Appendix

TL;DR

This paper develops a comprehensive spectral theory for mean values of automorphic L-functions, extending the fourth moment of the Riemann zeta-function with new methods and explicit reasoning.

Contribution

It introduces a unified, detailed approach to the spectral theory of automorphic L-functions, including a simplified geometric understanding of sum formulas involving Kloosterman sums.

Findings

Resolved the mean value problem for automorphic L-functions.

Provided a simplified, explicit account of Cogdell-Piatetski-Shapiro's method.

Extended the spectral theory beyond traditional approaches.

Abstract

The main aim of this article is to develop, in a fully detailed fashion, a {\bf unified} theory of the spectral theory of mean values of individual automorphic L-functions which is a natural extension of the fourth moment of the Riemann zeta-function but does not admit any analogous argument and requires a genuinely new method. Thus we first develop a relatively self-contained account of the theory of automorphic representations, especially highlighting the Kirillov model, with which we resolve the problem on the mean value of those L-functions. As another reward, we gain a geometrical understanding of sum formulas involving Kloosterman sums, which is in fact a considerably simplified account of Cogdell-Piatetski-Shapiro's method. Our reasoning is quite explicit in contrast to theirs.