In Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes

Couverture
American Mathematical Soc., 2008 - 558 pages
Formulated in 1859, the Riemann Hypothesis is the most celebrated and multifaceted open problem in mathematics. In essence, it states that the primes are distributed as harmoniously as possible--or, equivalently, that the Riemann zeros are located on a single vertical line, called the critical line. In this book, the author proposes a new approach to understand and possibly solve the Riemann Hypothesis. His reformulation builds upon earlier (joint) work on complex fractal dimensions and the vibrations of fractal strings, combined with string theory and noncommutative geometry. Accordingly, it relies on the new notion of a fractal membrane or quantized fractal string, along with the modular flow on the associated moduli space of fractal membranes. Conjecturally, under the action of the modular flow, the spacetime geometries become increasingly symmetric and crystal-like, hence, arithmetic. Correspondingly, the zeros of the associated zeta functions eventually condense onto the critical line, towards which they are attracted, thereby explaining why the Riemann Hypothesis must be true. Written with a diverse audience in mind, this unique book is suitable for graduate students, experts and nonexperts alike, with an interest in number theory, analysis, dynamical systems, arithmetic, fractal or noncommutative geometry, and mathematical or theoretical physics.
 

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Pages sélectionnées

Table des matières

Introduction
1
String Theory on a Circle and TDuality Analogy with the Riemann Zeta Function
21
Fractal Strings and Fractal Membranes
89
Noncommutative Models of Fractal Strings Fractal Membranes and Beyond
155
Towards an Arithmetic Site Moduli Spaces Fractal Strings and Membranes
197
Vertex Algebras
315
The Weil Conjectures and the Riemann Hypothesis
325
The Poisson Summation Formula with Applications
347
The Selberg Class of Zeta Functions
389
The Noncommutative Space of Penrose Tilings and Quasicrystals
411
Bibliography
453
Conventions
491
Subject Index
503
Author Index
551
Droits d'auteur

Expressions et termes fréquents

Fréquemment cités

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Page 478 - DD OSHEROFF, RC RICHARDSON, AND DM LEE, Evidence for a new phase of solid 3He, Phys.
Page 486 - M. Tomita, Standard forms of von Neumann algebras, Fifth Functional Analysis Symposium of the Math. Soc. of Japan, Sendai, 1967.

Informations bibliographiques