Group items tagged

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

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).

Among the modern methods used to study prime numbers, the ‘sieve’ has been one of the most efficient. Originally conceived by Linnik in 1941, the ‘large sieve’ has developed extensively since the 1960s, with a recent realisation that the underlying principles were capable of applications going well beyond prime number theory. This book develops a general form of sieve inequality, and describes its varied applications, including the study of families of zeta functions of algebraic curves over finite fields; arithmetic properties of characteristic polynomials of random unimodular matrices; homological properties of random 3-manifolds; and the average number of primes dividing the denominators of rational points on elliptic curves. Also covered in detail are the tools of harmonic analysis used to implement the forms of the large sieve inequality, including the Riemann Hypothesis over finite fields, and Property (T) or Property (tau) for discrete groups.• Explores new and surprising applications of the large sieve method, an important technique of analytic number theory • Presents applications in fields as wide ranging as topology, probability, arithmetic geometry and discrete group theory • Motivated, clear and self-contained discussions introduce readers to a technique previously confined to one fieldContentsPreface; Prerequisites and notation; 1. Introduction; 2. The principle of the large sieve; 3. Group and conjugacy sieves; 4. Elementary and classical examples; 5. Degrees of representations of finite groups; 6. Probabilistic sieves; 7. Sieving in discrete groups; 8. Sieving for Frobenius over finite fields; Appendix A. Small sieves; Appendix B. Local density computations over finite fields; Appendix C. Representation theory; Appendix D. Property (T) and Property (τ); Appendix E. Linear algebraic groups; Appendix F. Probability theory and random walks; Appendix G. Sums of multiplicative functions; Appendix H. Topology; Bibliography; Index.

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?

An alternate argument for the arithmetic large sieve inequality September 2008 This short note describes a very natural and well-motivated derivation of the "arithmetic" large sieve inequality from the dual of the analytic inequality, which avoids the usual trick of submultiplicativity of Gallagher. This is also described in a blog post.

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.

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.

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).

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.

We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system of representations arising from the Galois action on the torsion points of an abelian variety. The resulting upper bounds require explicit character sum calculations, with stronger results holding if one assumes the Generalized Riemann Hypothesis.

Are there infinitely many prime pairs with given even difference? Most mathematicians think so. Using a strong arithmetic hypothesis, Goldston, Pintz and Yildirim have recently shown that there are infinitely many pairs of primes differing by at most sixteen.There is extensive numerical support for the prime-pair conjecture (PPC) of Hardy and Littlewood [G.H. Hardy, J.E. Littlewood, Some problems of ‘partitio numerorum’. III: On the expression of a number as a sum of primes, Acta Math. 44 (1923) 1–70 (sec. 3)] on the asymptotic behavior of π2r(x), the number of prime pairs with p≤x. Assuming Riemann’s Hypothesis (RH), Montgomery and others have studied the pair-correlation of zeta’s complex zeros, indicating connections with the PPC. Using a Tauberian approach, the author shows that the PPC is equivalent to specific boundary behavior of a function involving zeta’s complex zeros. A certain hypothesis on equidistribution of prime pairs, or a speculative supplement to Montgomery’s work on pair-correlation, would imply that there is an abundance of prime pairs.

We describe a very general abstract form of sieve based on a large sieve inequality which generalizes both the classical sieve inequality of Montgomery (and its higher-dimensional variants), and our recent sieve for Frobenius over function fields. The general framework suggests new applications. We get some first results on the number of prime divisors of ``most'' elements of an elliptic divisibility sequence, and we develop in some detail ``probabilistic'' sieves for random walks on arithmetic groups, e.g., estimating the probability of finding a reducible characteristic polynomial at some step of a random walk on SL(n,Z). In addition to the sieve principle, the applications depend on bounds for a large sieve constant. To prove such bounds involves a variety of deep results, including Property (T) or expanding properties of Cayley graphs, and the Riemann Hypothesis over finite fields. It seems likely that this sieve can have further applications.

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.

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.