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

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

2008-10-16

arxiv.org/abs/0810.2847v1

Mathematics

Number Theory

45 pages. This is to be an addition to my book, the title of which is indicated in the title of the present work. The full tex

Scientific paper

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.

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