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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

For any and any non-exceptional modulus , we prove that, for large enough ( ), the interval contains a prime in any of the arithmetic progressions modulo . We apply this result to establish that every integer larger than is a sum of seven cubes.

Inverse Conjecture for the Gowers norm is false Authors: Shachar Lovett, Roy Meshulam, Alex Samorodnitsky (Submitted on 21 Nov 2007) Abstract: Let $p$ be a fixed prime number, and $N$ be a large integer. The 'Inverse Conjecture for the Gowers norm' states that if the "$d$-th Gowers norm" of a function $f:\F_p^N \to \F_p$ is non-negligible, that is larger than a constant independent of $N$, then $f$ can be non-trivially approximated by a degree $d-1$ polynomial. The conjecture is known to hold for $d=2,3$ and for any prime $p$. In this paper we show the conjecture to be false for $p=2$ and for $d = 4$, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao \cite{gt07}. Their analysis uses a modification of a Ramsey-type argument of Alon and Beigel \cite{ab} to show inapproximability of certain functions by low-degree polynomials. We observe that a combination of our results with the argument of Alon and Beigel implies the inverse conjecture to be false for any prime $p$, for $d = p^2$. Comments: 20 pages

Pseudo Random test of prime numbers Authors: Wang Liang; Huang Yan (Submitted on 18 Mar 2006) Abstract: The prime numbers look like a randomly chosen sequence of natural numbers, but there is still no strict theory to determine 'Randomness'. In these years, cryptography has developed a battery of statistical tests for randomness. In this paper, we just apply these methods to study the distribution of primes. Here the binary sequence constructed by second difference of primes is used as samples. We find this sequence can't reach all the 'random standard' of FIPS 140-1/2, but still show obvious random feature. The interesting self-similarity is also observed in this sequence. These results add the evidence that prime numbers is a chaos system.

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.

MATH 254A : Topics in Ergodic Theory Course description: Basic ergodic theorems (pointwise, mean, maximal) and recurrence theorems (Poincare, Khintchine, etc.)  Topological dynamics.  Structural theory of measure-preserving systems; characteristic factors.  Spectral theory of dynamical systems.  Multiple recurrence theorems (Furstenberg, etc.) and connections with additive combinatorics (e.g. Szemerédi’s theorem).  Orbits in homogeneous spaces, especially nilmanifolds; Ratner’s theorem.  Further topics as time allows may include joinings, dynamical entropy, return times theorems, arithmetic progressions in primes, and/or

        Instructor: Terence Tao, tao@math.ucla.edu, x64844, MS 6183

The connections between the zeros of the zeta-function and random matrix theory have become the most active and exciting threads of research in the hunt for the Riemann hypothesis. Rockmore devotes four chapters at the end of his book to various aspects of this research. He discusses the work of Sarnak and Katz on analogous results for function fields. He also discusses work of Tracy, Widom, and Deift that connects the distribution of eigenvalues of random matrices to properties of permutations. This chapter has the engaging title "God May Not Play Dice, but What About Cards?"

This course is an introduction to analytic number theory, including the use of zeta functions and L-functions to prove distribution results concerning prime numbers (e.g., the prime number theorem in arithmetic progressions).

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).

Topics in Harmonic Analysis and Ergodic Theory Joseph M. Rosenblatt, Alexander M. Stokolos, Ahmed I. Zayed ISBN: 0821842358 Paperback American Mathematical Society Usually despatched within 3 to 9 days

There are strong connections between harmonic analysis and ergodic theory. A recent example of this interaction is the proof of the spectacular result by Terence Tao and Ben Green that the set of prime numbers contains arbitrarily long arithmetic progressions. The breakthrough achieved by Tao and Green is attributed to applications of techniques from ergodic theory and harmonic analysis to problems in number theory.Articles in the present volume are based on talks delivered by plenary speakers at a conference on Harmonic Analysis and Ergodic Theory (DePaul University, Chicago, December 2-4, 2005). Of ten articles, four are devoted to ergodic theory and six to harmonic analysis, although some may fall in either category. The articles are grouped in two parts arranged by topics. Among the topics are ergodic averages, central limit theorems for random walks, Borel foliations, ergodic theory and low pass filters, data fitting using smooth surfaces, Nehari's theorem for a polydisk, uniqueness theorems for multi-dimensional trigonometric series, and Bellman and s-functions.In addition to articles on current research topics in harmonic analysis and ergodic theory, this book contains survey articles on convergence problems in ergodic theory and uniqueness problems on multi-dimensional trigonometric series.

We extend two well-known results in additive number theory, S\'ark\"ozy's theorem on square differences in dense sets and a theorem of Green on long arithmetic progressions in sumsets, to subsets of random sets of asymptotic density 0. Our proofs rely on a restriction-type Fourier analytic argument of Green and Green-Tao.