ErgodicPNT

Behrend's construction

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.

The paper itself is called “Hypergraph regularity and the multidimensional Szemerédi theorem.” At the bottom level, the basic idea of the paper is due to Ruzsa, Szemerédi and Rödl. Ruzsa and Szemerédi started the ball rolling with a short and very clever argument that showed that Szemerédi’s famous theorem on arithmetic progressions, in the case of progressions of length 3, could be deduced from Szemerédi’s almost as famous regularity lemma, a remarkable result that allows any graph to be partitioned into a bounded number of pieces, almost all of which “behave randomly.”

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 $d = p^2$.

Solving Mathematical Problems: A personal perspective. 2nd Edition, Terence Tao. Oxford University P ?The Journey of a thousand miles begins with one step? ? Lao Tzu Every so often, you come across a book that really stands out. I have recently been very fortunate to come across several such books, this being one of them. ?Solving mathematical problems? was written by Terence Tao when he was a 15 year old student and has now been slightly revised in this second edition. Like another of the reviewers at Amazon, I also came across this book after reading an article about Terence Tao winning the Fields medal (a bit like the Nobel prize for mathematics). Not only does it give a wonderful insight into the mind of a young Terence Tao, but also into the techniques used to elegantly solve some reasonably difficult problems, such as those posed as questions for the Maths Olympiad contests. [Terence competed in these challenges in his teens, winning bronze, silver and then gold.] Mathematical researchers are not always great e\ucators. Thankfully, Prof. Tao is.Mainly assuming only basic high-school pure mathematics, worked solutions to the problems are clearly and expertly described. Not only does he solve the problems but he also examines the steps, false starts and other solution possibilities that are part of the general approach to problem solving. I was only slightly disappointed that there were a handful of corrections in this second edition (available at Prof Tao?s blog here); one or two could perplex an unwary reader who might expect the work to be flawless. If you have an interest in mathematics, either as a high school student or a hobbyist, I would highly recommended reading this book. In the preface, Prof Tao remarks that if he wrote a book on the subject of competition problem-solving now, it would very different; now that is definitely a book I would like to read!….

Made to Stick: Why Some Ideas Survive and Others Die by Chip and Dan Heath helps us understand why our users (or our coworkers) can repeat the latest web hoax, but can’t remember anything about our projects. What we need to do is to create “sticky messages.” Sticky messages are not necessarily creative messages. In fact, there is formula that the brothers Heath have discovered that will help us to create sticky, memorable messages. That formula is:

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

Quadratic Uniformity of the Mobius Function Authors: Ben Green, Terence Tao (Submitted on 4 Jun 2006 (v1), last revised 22 Sep 2007 (this version, v2)) Abstract: This paper is a part of our programme to generalise the Hardy-Littlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear Equations in Primes [LEP], which accompanies this submission. In particular, the results of this paper may be used, together with the machinery of [LEP], to establish an asymptotic for the number of four-term progressions p_1 < p_2 < p_3 < p_4 <= N of primes, and more generally any problem counting prime points inside a ``non-degenerate'' affine lattice of codimension at most 2. The main result of this paper is a proof of the Mobius and Nilsequences Conjecture for 1 and 2-step nilsequences. This conjecture is introduced in [LEP] and amounts to showing that if G/\Gamma is an s-step nilmanifold, s <= 2, if F : G/\Gamma -> [-1,1] is a Lipschitz function, and if T_g : G/\Gamma -> G/\Gamma is the action of g \in G on G/\Gamma, then the Mobius function \mu(n) is orthogonal to the sequence F(T_g^n x) in a fairly strong sense, uniformly in g and x in G/\Gamma. This can be viewed as a ``quadratic'' generalisation of an exponential sum estimate of Davenport, and is proven by the following the methods of Vinogradov and Vaughan.

However, it turns out that if one of the sets, say A, is sufficiently “uniform” or “pseudorandom”, then one can always solve this Goldbach-type problem, regardless of what the other two sets are. This type of fact is often established by Fourier-analytic means (or by closely related techniques, such as spectral theory), but let me give a heuristic combinatorial argument to indicate why one would expect this type of phenomenon to occur.

quares Primes Lagrange’s four square theorem: For every positive integer N, there exists a pattern in of the form . Vinogradov’s theorem: For every sufficiently large integer N, there exists a pattern in of the form . Fermat’s two square theorem: For every prime number , there exists a pattern in of the form . Even Goldbach conjecture: For every even number , there exists a pattern in of the form . Fermat’s four square theorem: There does not exist any pattern in of the form with . Green-Tao theorem: For any , there exist infinitely many patterns in of the form with . Pell’s equation: There are infinitely many patterns in of the form . Sophie Germain conjecture: There are infinitely many patterns in of the form .

The primary aim of this book is to create situations in which the zeta function, or other L-functions, will appear in spectral-theoretic questions. A secondary aim is to connect pseudo-differential analysis, or quantization theory, to analytic number theory. Both are attained through the analysis of operators on functions on the line by means of their diagonal matrix elements against families of arithmetic coherent states: these are families of discretely supported measures on the line, transforming in specific ways under the part of the metaplectic representation or, more generally, representations from the discrete series of SL(2,R), lying above an arithmetic group such as SL(2,Z).

2 July, 2008 at 6:28 pm Terence Tao It unfortunately seems that the decomposition claimed in equation (6.9) on page 20 of that paper is, in fact, impossible; it would endow the function h (which is holding the arithmetical information about the primes) with an extremely strong dilation symmetry which it does not actually obey. It seems that the author was relying on this symmetry to make the adelic Fourier transform far more powerful than it really ought to be for this problem.

3 July, 2008 at 3:41 am Gergely Harcos I also have some (perhaps milder) troubles with the proof. It seems to me as if Li had treated the Dirac delta on L^2(A) as a function. For example, the first 5 lines of page 28 make little sense to me. Am I missing something here?

4 July, 2008 at 5:15 am Lior Silberman The function defined on page 20 does have a strong dilation symmetry: it is invariant by multiplication by ideles of norm one (since it is merely a function of the norm of ). In particular, it is invariant under multiplication by elements of . I’m probably missing something here. Probably the subtlety is in passing from integration over the nice space of idele classes to the singular space . The topologies on the spaces of adeles and ideles are quite different. There is a formal error in Theorem 3.1 which doesn’t affect the paper: the distribution discussed is not unique. A distribution supported at a point is a sum of derivatives of the delta distribution. Clearly there exist many such with a given special value of the Fourier transform. There is also something odd about this paper: nowhere is it pointed out what is the new contribution of the paper. Specifically, what is the new insight about number theory?

In the second lecture (based on Gowers’s joint work with Julia Wolf) we were introduced to decomposition theorems. A decomposition theorem for the norm can be stated as follows: if is a function (on either or ) with , there is a decomposition , where are “generalized quadratic phase functions” and are error terms with and small. This can be deduced from the inverse theorem of Green-Tao; in fact a similar statement was already implicit in their work, based on the energy increment argument. Tim presented a different approach to deducing decomposition theorems from inverse theorems, based on functional-analytic arguments involving the geometry of normed spaces (specifically, a variant of the Hahn-Banach theorem).

This can be applied to the question of counting solutions to systems of linear equations in sets. Let’s say that we are interested in finding sensible conditions under which a set will have the “statistically correct” number of solutions to a system of linear equations. For instance, if it is 4-term arithmetic progressions that we are concerned with, then uniformity is sufficient (and, in general, necessary). Green and Tao prove a more general result of this type: they define the complexity of a system of linear forms, and prove that systems of complexity are controlled by norms.

Gowers and Wolf, however, do not stop there. Suppose that, instead of 4-term progressions, we are interested in configurations of the form, say, . The complexity of this system in the sense of Green-Tao is 2, hence a set uniform in the norm will contain the “right” number of such configurations. Gowers and Wolf, however, can prove that uniformity already guarantees the same conclusion! The difference between the two examples? The squares are linearly dependent, whereas are not. Gowers and Wolf prove that such “square independence” is in fact both sufficient and necessary for a system of complexity 2 to be controlled by the $U^2$ norm. The proof is based on the decomposition theorem described earlier.

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Many areas of active research within the broad field of number theory relate to properties of polynomials, and this volume displays the most recent and most interesting work on this theme. The 2006 Number Theory and Polynomials workshop in Bristol drew together international researchers with a variety of number-theoretic interests, and the book’s contents reflect the quality of the meeting. Topics covered include recent work on the Schur-Siegel-Smyth trace problem, Mahler measure and its generalisations, the merit factor problem, Barker sequences, K3-surfaces, self-inversive polynomials, Newman’s inequality, algorithms for sparse polynomials, the integer transfinite diameter, divisors of polynomials, non-linear recurrence sequences, polynomial ergodic averages, and the Hansen-Mullen primitivity conjecture. With surveys and expository articles presenting the latest research, this volume is essential for graduates and researchers looking for a snapshot of current progress in polynomials and number theory.• An invaluable resource to both students and experts in this area, with survey articles on the most important topics in the field • Expository articles introduce graduate students to some problems of active interest • The inclusion of new results from leading experts in the field provides a snapshot of current progress

From their inception, Siegel modular forms have been studied extensively because of their significance in both automorphic functions in several complex variables and number theory. The comprehensive theory of automorphic forms to subgroups of algebraic groups and the arithmetical theory of modular forms illustrate these two aspects in an illuminating manner. The author’s aim is to present a straightforward and easily accessible survey of the main ideas of the theory at an elementary level, providing a sound basis from which the reader can study advanced works and undertake original research. This book is based on lectures given by the author for a number of years and is intended for a one-semester graduate course, though it can also be used profitably for self-study. The only prerequisites are a basic knowledge of algebra, number theory and complex analysis.Contents

Preface; 1. The modular group; 2. Basic facts on modular forms; 3. Large weights; 4. Small weights; 5. Modular functions; 6. Dirichlet series; Bibliography; Index.