Spectral Theory of the Riemann Zeta-Function

by Yoichi Motohashi

• ISBN13: 9780521445207
• ISBN10: 0521445205
• Format: Hardcover
• Copyright: 1997-10-13
• Publisher: Cambridge University Press

Summary

The Riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. In this book, based on his own research, Professor Motohashi shows that the function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the zeta function itself. The story starts with an elementary but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. This is achieved by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory as well. These ideas are then utilized to unveil an image of the zeta-function, first perceived by the author, revealing it to be the main gem of a necklace composed of all automorphic L-functions. In this book, readers will find a detailed account of one of the most fascinating stories in the development of number theory, namely the fusion of two main fields in mathematics that were previously studied separately.

Table of Contents

Preface	vii	(1)
Convention and assumed background	viii
1 Non-Euclidean harmonics	1	(42)
1.1 Basic concepts	1	(16)
1.2 Resolvent kernel	17	(5)
1.3 Interated kernel	22	(10)
1.4 Spectral resolution	32	(8)
1.5 Notes for Chapter 1	40	(3)
2 Trace formulas	43	(54)
2.1 Basic identities	43	(6)
2.2 Holomorphic forms	49	(5)
2.3 Spectral means of Mass-Fourier coefficients	54	(7)
2.4 Sums of Kloosterman sums	61	(6)
2.5 Opposite-sign case	67	(5)
2.6 Alternative argument	72	(11)
2.7 A flight	83	(7)
2.8 Notes for Chapter 2	90	(7)
3 Automorphic L-functions	97	(47)
3.1 Hecke operators	97	(7)
3.2 L-functions	104	(7)
3.3 Bilinear forms of L-functions	111	(10)
3.4 Spectral mean values	121	(10)
3.5 Spectral large sieve	131	(8)
3.6 Notes for Chapter 3	139	(5)
4 An explicit formula	144	(41)
4.1 A prototype	145	(5)
4.2 Non-diagonals	150	(3)
4.3 Reduction to Kloosterman sums	153	(5)
4.4 Spectral expansion	158	(6)
4.5 Crystallization	164	(2)
4.6 Analytic continuation	166	(5)
4.7 Explict formula	171	(7)
4.8 Notes for Chapter 4	178	(7)
5 Asymptotics	185	(36)
5.1 Local explicit formula	185	(14)
5.2 Global bounds	199	(6)
5.3 Eigen-peaks	205	(7)
5.4 Notes for Chapter 5, and epilogue	212	(9)
References	221	(4)
Author Index	225	(2)
Subject index	227