ErgodicPNT

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.

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).

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).

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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?"

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

wo researchers from the University of Bristol exhibited the first example of a third degree transcendental L-function.

"This work was made possible by a combination of theoretical advances and the power of modern computers." During his lecture, Bian reported that it took approximately 10,000 hours of computer time to produce his initial results.

We show that if A is a subset of Z_4^n containing no three term arithmetic progression in which all the elements are distinct then |A|=o(4^n/n).

Gergely Harcos The precise bound was first proved by Wigert in 1906 using the prime number theorem, while Ramanujan in 1914 observed its elementary character. In fact we can prove the inequality even without knowing unique factorization! All we need to know is that and imply . This property implies as one can inject the set of divisors of into the set of pairs formed of a divisor of and a divisor of : to assign the pair . Once we know we can see for any positive integer that . It follows that , whence also . Now the second exponent changes by a factor less than 2 whenever is increased by 1, so we can certainly find a with . This choice furnishes Wigert’s estimate upon observing that .

Checking the Odd Goldbach Conjecture Up to 1020

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.

Roth’s theorem on progressions revisited

The Riemann Hypothesis is reformulated as statements about eigenvalues of some matrices entries of which are defined via Taylor coefficient of the zeta function. These eigenvalues demonstrate interesting visual patterns allowing one to state a number of conjectures.

A new norm related to the Gowers U^3 norm Add to your list(s) Download to your calendar using vCal Pablo Candela Pokorna Monday 16 February 2009, 16:00-17:00 MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB. If you have a question about this talk, please contact Anton Evseev. The uniformity norms (or U^d norms, for d>1 a positive integer) were introduced about ten years ago by Gowers in his effective proof of Szemerédi’s theorem, and have played an important role in arithmetic combinatorics ever since. The U^2 norm is naturally related to Fourier analysis, and a very active trend in current research aims to develop an analogue of Fourier analysis for each U^d norm with d>2. The body of results of this research for d=3 is known as quadratic Fourier analysis. After an introduction to this area we will consider a new norm related to the U^3 norm, and discuss some of its applications in quadratic Fourier analysis, including a strengthening of a central theorem of Green and Tao (the inverse theorem for the U^3 norm), and how this stronger version of the theorem can be used to give a new proof of a recent decomposition-theorem of Gowers and Wolf. This talk is part of the Junior Algebra/Combinatorics/Number Theory seminar series.

There is an amusing way to interpret the conjecture (using the close relationship between nilsequences and bracket polynomials) as an assertion of the pseudorandomness of the Liouville function from a computational complexity perspective.   

Number theory : A Computational Introduction to Number Theory and Algebra - Victor Shoup A Concise Introduction to the Theory of Numbers- Baker A. A Course in Arithmetic (graduate level) - J. Serre A course in computational algebraic number theory - Cohen H. A Course in Number Theory and Cryptography 2 ed - Neal Koblitz A Course In Number Theory And Cryptography 2Ed - Koblitz N Advanced Number Theory - Cohn Algebra and number theory - Baker A. Algebraic Groups and Number Theory - Platonov & Rapinchuk Algebraic Number Theory - IYANAGA ALGEBRAIC NUMBER THEORY - MILNE Algorithmic Methods In Algebra And Number Theory - Pohst M Algorithmic number theory - Cohen H. Algorithmic number theory, vol. 1 Efficient algorithms - Bach E., Shallit J. An Explicit Approach To Elementary Number Theory - stein An Introduction to Conformal Field Theory [jnl article] - M. Gaberdiel AN INTRODUCTION TO THE THEORY OF NUMBERS - hardy & wright An Introduction to the Theory of Numbers - Leo Moser An introduction to the theory of numbers 5ed - Niven I., Zuckerman H.S., Montgomery H.L. Analytic number theory - Iwaniec H.,Kowalski E. Analytic Number Theory - Newman D.J. Analytic Number Theory- Jia & Matsumoto Arithmetic Theory of Elliptic Curves - J. Coates Computational Algebraic Number Theory - Pohst M E Computational excursions in analysis and number theory - Borwein P.

Only Problems Not Solutions - F. Smarandache Prime Numbers The Most Mysterious Figures in Math - D. Wells Problems In Algebraic Number Theory 2Ed - Murty M , Esmonde J SOlved and unsolved problems in Number Theory - Daniel Shanks Surfing on the Ocean of Numbers - H. Ibstedt Survey Of Diophantine Geometry - Serge Lang The elements of the theory of algebraic numbers - Hilbert.djv The Foundations of Arithmetic 2nd ed. revised - G. Frege The New Book Of Prime Number Records 3rd ed. - P. Ribenboim The Theory of algebraic numbers sec ed - Pollard H., Diamond H.G. the theory of functions and sets of natural numbers - Odifreddi, P Three Pearls of Number Theory - Khinchin Transcendental number theory - Baker A. Unsolved Problems In Number Theory 2 Ed - R K Guy.djv

Introduction to Modern Number Theory Fundamental Problems, Ideas and Theories 2nd Edition - Manin I., Panchishkin A Introduction to p-adic numbers and valuation theory- Bachman G. Introduction to the Theory of Numbers 4th ed. - G. Hardy, E. Wright Lectures on topics in algebraic number theory - Ghorpade Mainly Natural Numbers - Studies on Sequences - H. Ibstedt Math. problems and proofs combinatorics, number theory and geometry - B. Kisacanin Mathematical Problems And Proofs Combinatorics, Number Theory, and Geometry - Kluwer Academic My Numbers, My Friends - Popular Lectures on Number Theory My Numbers,My Friends Popular Lectures On Number Theory - Ribenboim Number Theory - Z.Borevitch, I. Shafarevich Number theory for beginners - Weil A. Number theory for computing - Yan S Y. Numerical Mathematics - A. Quarteroni, A. Sacco, F. Saleri Numerical Methods for Engineers and Scientists 2nd ed. - J. Hoffman Numerical Optimization - J. Nocedal, S. Wright Numerical Recipes in C - The Art Of Scientific Computing 2nd ed. Numerical Recipes in Fortran 77 2nd ed. Vol 1 Old And New Problems And Results In Combinatorial Number Theory - Erdos, P.&Graham, R.L

For instance, the question of whether is equidistributed mod 1 is an old unsolved problem, equivalent to asking whether is normal base 10.

For instance, the question of whether is equidistributed mod 1 is an old unsolved problem, equivalent to asking whether is normal base 10.

[Incidentally, regarding the interactions between physics and number theory: physical intuition has proven to be quite useful in making accurate predictions about many mathematical objects, such as the distribution of zeroes of the Riemann zeta function, but has been significantly less useful in generating rigorous proofs of these predictions. In number theory, our ability to make accurate predictions on anything relating to the primes (or related objects) is now remarkably good, but our ability to actually prove these predictions rigorously lags behind quite significantly. So I doubt that the key to further rigorous progress on these problems lies with physics.]