2008;
558 pp;
Hardcover

MSC: Primary 11; 28; 37; 46; 58; 81;

Print ISBN: 978-0-8218-4222-5

Product Code: MBK/51

List Price: $86.00

AMS Member Price: $68.80

MAA Member Price: $77.40

**Electronic ISBN: 978-1-4704-2482-4
Product Code: MBK/51.E**

List Price: $86.00

AMS Member Price: $68.80

MAA Member Price: $77.40

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#### Supplemental Materials

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

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*Michel L. Lapidus*

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.

#### Readership

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

#### Reviews & Endorsements

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

# Table of Contents

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

- Cover Cover11
- Title page iii4
- Dedication v6
- Quotes vii8
- Contents ix10
- Preface xiii14
- Acknowledgments xvii18
- Credits xxiii24
- Overview xxv26
- About the cover xxix30
- Introduction 132
- String theory on a circle and T-duality: Analogy with the Riemann zeta function 2152
- Fractal strings and fractal membranes 89120
- Noncommutative models of fractal strings: Fractal membranes and beyond 155186
- Towards an ‘arithmetic site’: Moduli spaces of fractal strings and membranes 197228
- Appendix A. Vertex algebras 315346
- Appendix B. The Weil conjectures and the Riemann hypothesis 325356
- Appendix C. The Poisson summation formula, with applications 347378
- Appendix D. Generalized primes and Beurling zeta functions 373404
- Appendix E. The Selberg class of zeta functions 389420
- Appendix F. The noncommutative space of Penrose tilings and quasicrystals 411442
- Bibliography 453484
- Conventions 491522
- Index of symbols 493524
- Subject index 503534
- Author index 551582
- Back Cover Back Cover1590