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

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

String Theory on a Circle and TDuality Analogy with the Riemann Zeta Function
Fractal Strings and Fractal Membranes
Noncommutative Models of Fractal Strings Fractal Membranes and Beyond
Towards an Arithmetic Site Moduli Spaces Fractal Strings and Membranes
Vertex Algebras
The Weil Conjectures and the Riemann Hypothesis
The Poisson Summation Formula with Applications
The Selberg Class of Zeta Functions
The Noncommutative Space of Penrose Tilings and Quasicrystals
Subject Index
Author Index
Droits d'auteur

Expressions et termes fréquents

Fréquemment cités

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Informations bibliographiques