Hardcover ISBN:  9780821842225 
Product Code:  MBK/51 
List Price:  $92.00 
MAA Member Price:  $82.80 
AMS Member Price:  $73.60 
Electronic ISBN:  9781470424824 
Product Code:  MBK/51.E 
List Price:  $86.00 
MAA Member Price:  $77.40 
AMS Member Price:  $68.80 

Book Details2008; 558 ppMSC: Primary 11; 28; 37; 46; 58; 81;
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 crystallike, 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.ReadershipGraduate students and research mathematicians interested in number theory, noncommutative geometry, and physics.

Table of Contents

Chapters

1. Introduction

2. String theory on a circle and Tduality: Analogy with the Riemann zeta function

3. Fractal strings and fractal membranes

4. Noncommutative models of fractal strings: Fractal membranes and beyond

5. Towards an ‘arithmetic site’: Moduli spaces of fractal strings and membranes

6. Vertex algebras

7. The Weil conjectures and the Riemann hypothesis

8. The Poisson summation formula, with applications

9. Generalized primes and Beurling zeta functions

10. The Selberg class of zeta functions

11. The noncommutative space of Penrose tilings and quasicrystals


Additional Material

Reviews

The author and the AMS have done an excellent job in the production of the book. ... Overall, this is a very interesting and very unconventional book.
Mathematicial Reviews


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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 crystallike, 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.
Graduate students and research mathematicians interested in number theory, noncommutative geometry, and physics.

Chapters

1. Introduction

2. String theory on a circle and Tduality: Analogy with the Riemann zeta function

3. Fractal strings and fractal membranes

4. Noncommutative models of fractal strings: Fractal membranes and beyond

5. Towards an ‘arithmetic site’: Moduli spaces of fractal strings and membranes

6. Vertex algebras

7. The Weil conjectures and the Riemann hypothesis

8. The Poisson summation formula, with applications

9. Generalized primes and Beurling zeta functions

10. The Selberg class of zeta functions

11. The noncommutative space of Penrose tilings and quasicrystals

The author and the AMS have done an excellent job in the production of the book. ... Overall, this is a very interesting and very unconventional book.
Mathematicial Reviews