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The Prime Numbers and Their Distribution User Rating: / 5 PoorBest  Written by Giulia Biagini    Sunday, 14 January 2007 Basic Information Title: The Prime Numbers and Their Distribution Authors: Gérald Tenenbaum and Michel Mendès France Paperback: 115 pages Publisher: American Mathematical Society (May 2000) Language: English ISBN-10: 0821816470 ISBN-13: 978-0821816479

This book gives a general and pleasing overview on many topics about the distribution of prime numbers. Its goal is to provide insights of different nature on that theme and this is performed through the illustration of conjectures, methods, results and even (very concise) proofs.   The volume is divided into five chapters, they are: Genesis: from Euclid to Chebyshev; The Riemann Zeta Function; Stochastic Distribution of Prime Numbers; An Elementary Proof of the Prime Number Theorem; The Major Conjectures. All of them are almost independent one to another, so you may skip the ones you are not interested in wIthout any problem. The first one consists of

In recent years the application of random matrix techniques to analytic number theory has been responsible for major advances in this area of mathematics. As a consequence it has created a new and rapidly developing area of research. The aim of this book is to provide the necessary grounding both in relevant aspects of number theory and techniques of random matrix theory, as well as to inform the reader of what progress has been made when these two apparently disparate subjects meet. This volume of proceedings is addressed to graduate students and other researchers in both pure mathematics and theoretical physics. The contributing authors, who are among the world leading experts in this area, have taken care to write self-contained lectures on subjects chosen to produce a coherent volume.• Self-contained lectures by world-leading experts in the field • The volume is integrated, indexed and cross-referenced • This title covers the most important and recent advances in the subjectContents1. Introduction; 2. Prime number theory and the Riemann zeta-function; 3. Notes on pair correlation of zeros and prime numbers; 4. Notes on eigenvalue distributions for the classical compact groups; 5. Compound nucleus resonances, random matrices and quantum chaos; 6. Families of L-functions and 1-level densities; 7. Basic analytic number theory; 8. Applications of mean value theorems to the theory of the Riemann zeta function; 9. L-functions and the characteristic polynomials of random matrices; 10. Mock gaussian behaviour; 11. Some specimens of L-functions; 12. Computational methods and experiments in analytic number theory.

References: Cipra, B. 1998. A prime case of chaos. In What's Happening in the Mathematical Sciences, Vol. 4. Providence, R.I.: American Mathematical Society. (Available at http://www.ams.org/new-in-math/happening.html.) ______. 1996. Prime formula weds number theory and quantum physics. Science 274(Dec. 20):2014. Davis, P.J., and R. Hersch. 1981. The Mathematical Experience. New York: Viking Penguin. Katz, N.M., and P. Sarnak. 1999. Zeroes of zeta functions and symmetry. Bulletin of the American Mathematical Society 36(January):1. Peterson, I. 1995. Cavities of chaos. Science News 147(April 29):264. Richards, I. 1978. Number theory. In Mathematics Today: Twelve Informal Essays. L.A. Steen, ed. New York: Springer-Verlag. Peter Sarnak's lecture on random matrix models in number theory and quantum mechanics is available at http://www.msri.org/publications/video/fall98/mandm.html. Andrew Odlyzko's Web page at http://www.research.att.com/~amo/ features computations of the zeros of the zeta function.

The Riemann hypothesis was first proposed in 1859 by the German mathematician Georg Friedrich Bernhard Riemann (1826-1866). It concerns the so-called zeta function, which encodes a great deal of information about the seemingly haphazard distribution of prime numbers among the integers (see The Mark of Zeta, June 19, 1999).