Download or read book Area Lattice Points and Exponential Sums written by M. N. Huxley and published by Clarendon Press. This book was released on 1996-06-13 with total page 510 pages. Available in PDF, EPUB and Kindle. Book excerpt: In analytic number theory a large number of problems can be "reduced" to problems involving the estimation of exponential sums in one or several variables. This book is a thorough treatment of the developments arising from the method developed by Bombieri and Iwaniec in 1986 for estimating the Riemann zeta function on the line *s = 1/2. Huxley and his coworkers (mostly Huxley) have taken this method and vastly extended and improved it. The powerful techniques presented here go considerably beyond older methods for estimating exponential sums such as van de Corput's method. The potential for the method is far from being exhausted, and there is considerable motivation for other researchers to try to master this subject. However, anyone currently trying to learn all of this material has the formidable task of wading through numerous papers in the literature. This book simplifies that task by presenting all of the relevant literature and a good part of the background in one package. The audience for the book will be mathematics graduate students and faculties with a research interest in analytic theory; more specifically, those with an interest in exponential sum methods. The book is self-contained; any graduate student with a one semester course in analytic number theory should have a more than sufficient background.
Download or read book Area Lattice Points and Exponential Sums written by Martin Neil Huxley and published by . This book was released on 1996 with total page 494 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Exponential Sums and the Riemann Zeta Function V written by Martin Neil Huxley and published by . This book was released on 2005 with total page 41 pages. Available in PDF, EPUB and Kindle. Book excerpt: A Van der Corput exponential sum is S = exp (2 i f(m)) where m has size M, the function f(x) has size T and = (log M) / log T 1. There are different bounds for S in different ranges for [greek letter alpha]. In the middle range where is near 1/over 2, S = [square root of MT subscript theta plus c]. This bounds the exponent of growth of the Riemann zeta function on its critical line Re s = 1/over 2. Van der Corput used an iteration which changed at each step. The Bombieri-Iwaniec method, whilst still based on mean squares, introduces number-theoretic ideas and problems. The Second Spacing Problem is to count the number of resonances between short intervals of the sum, when two arcs of the graph of y = f(x) coincide approximately after an automorphism of the integer lattice. In the previous paper in this series [Proc. London Math. Soc. (3) 66 (1993) 1-40] and the monograph Area, lattice points, and exponential sums we saw that coincidence implies that there is an integer point close to some 'resonance curve', one of a family of curves in some dual space, now calculated accurately in the paper 'Resonance curves in the Bombieri-Iwaniec method', which is to appear in Funct. Approx. Comment. Math. We turn the whole Bombieri-Iwaniec method into an axiomatised step: an upper bound for the number of integer points close to a plane curve gives a bound in the Second Spacing Problem, and a small improvement in the bound for S. Ends and cusps of resonance curves are treated separately. Bounds for sums of type S lead to bounds for integer points close to curves, and another branching iteration. Luckily Swinnerton-Dyer's method is stronger. We improve from 0.156140... in the previous paper and monograph to 0.156098.... In fact (32/205 +, 269/410 +) is an exponent pair for every 0. 2000 Mathematics Subject Classification 11L07 (primary), 11M06, 11P21, 11J54 (secondary).
Download or read book Lattice Points written by Ekkehard Krätzel and published by Springer Science & Business Media. This book was released on 1989-03-31 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to a special problem of number theory, that is the estimation of the number of lattice points in large closed domains of ordinary Euclidean spaces. Circle and sphere problems, Dirichlet's divisor problem, the distribution of powerful numbers, and finite Abelian groups are also investigated. The object of this book is to acquaint the reader with the fundamental results and methods, so that follow up with the original papers is possible.
Download or read book Number Theory written by R.P. Bambah and published by Birkhäuser. This book was released on 2012-12-06 with total page 525 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Indian National Science Academy on the occasion ofthe Golden Jubilee Celebration (Fifty years of India's Independence) decided to publish a number of monographs on the selected fields. The editorial board of INS A invited us to prepare a special monograph in Number Theory. In reponse to this assignment, we invited several eminent Number Theorists to contribute expository/research articles for this monograph on Number Theory. Al though some ofthose invited, due to other preoccupations-could not respond positively to our invitation, we did receive fairly encouraging response from many eminent and creative number theorists throughout the world. These articles are presented herewith in a logical order. We are grateful to all those mathematicians who have sent us their articles. We hope that this monograph will have a significant impact on further development in this subject. R. P. Bambah v. C. Dumir R. J. Hans-Gill A Centennial History of the Prime Number Theorem Tom M. Apostol The Prime Number Theorem Among the thousands of discoveries made by mathematicians over the centuries, some stand out as significant landmarks. One of these is the prime number theorem, which describes the asymptotic distribution of prime numbers. It can be stated in various equivalent forms, two of which are: x (I) K(X) '" -I - as x --+ 00, ogx and Pn '" n log n as n --+ 00. (2) In (1), K(X) denotes the number of primes P ::s x for any x > O.
Download or read book Surveys in Number Theory written by Bruce Berndt and published by CRC Press. This book was released on 2002-11-20 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume, based on fourteen papers from the Millennial Conference on Number Theory, represents surveys of topics in number theory and provides an outlook into the future of number theory research. It serves as an inspiration to graduate students and as a reference for research mathematicians.
Download or read book Number Theory written by Matti Jutila and published by Walter de Gruyter. This book was released on 2014-01-02 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: These Proceedings contain 22 refereed research and survey articles based on lectures given at the Turku Symposium on Number Theory in Memory of Kustaa Inkeri, held in Turku, Finland, from May 31 to June 4, 1999. The subject of the symposium was number theory in a broad sense with an emphasis on recent advances and modern methods. The topics covered in this volume include various questions in elementary number theory, new developments in classical Diophantine problems - in particular of the Fermat and Catalan type, the ABC-conjecture, arithmetic algebraic geometry, elliptic curves, Diophantine approximations, Abelian fields, exponential sums, sieve methods, box splines, the Riemann zeta-function and other Dirichlet series, and the spectral theory of automorphic functions with its arithmetical applications.
Download or read book Number Theory for the Millennium II written by Bruce Berndt and published by CRC Press. This book was released on 2024-07-31 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: Building on the tradition of an outstanding series of conferences at the University of Illinois at Urbana-Champaign, the organizers attracted an international group of scholars to open the new Millennium with a conference that reviewed the current state of number theory research and pointed to future directions in the field. The conference was the largest general number theory conference in recent history, featuring a total of 159 talks, with the plenary lectures given by George Andrews, Jean Bourgain, Kevin Ford, Ron Graham, Andrew Granville, Roger Heath-Brown, Christopher Hooley, Winnie Li, Kumar Murty, Mel Nathanson, Ken Ono, Carl Pomerance, Bjorn Poonen, Wolfgang Schmidt, Chris Skinner, K. Soundararajan, Robert Tijdeman, Robert Vaughan, and Hugh Williams. The Proceedings Volumes of the conference review some of the major number theory achievements of this century and to chart some of the directions in which the subject will be heading during the new century. These volumes will serve as a useful reference to researchers in the area and an introduction to topics of current interest in number theory for a general audience in mathematics.
Download or read book Unsolved Problems in Number Theory written by Richard Guy and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 455 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics is kept alive by the appearance of new, unsolved problems. This book provides a steady supply of easily understood, if not easily solved, problems that can be considered in varying depths by mathematicians at all levels of mathematical maturity. This new edition features lists of references to OEIS, Neal Sloane’s Online Encyclopedia of Integer Sequences, at the end of several of the sections.
Download or read book Metaharmonic Lattice Point Theory written by Willi Freeden and published by CRC Press. This book was released on 2011-05-09 with total page 467 pages. Available in PDF, EPUB and Kindle. Book excerpt: Metaharmonic Lattice Point Theory covers interrelated methods and tools of spherically oriented geomathematics and periodically reflected analytic number theory. The book establishes multi-dimensional Euler and Poisson summation formulas corresponding to elliptic operators for the adaptive determination and calculation of formulas and identities of
Download or read book Introduction to Analytic and Probabilistic Number Theory written by Gérald Tenenbaum and published by American Mathematical Society. This book was released on 2024-06-26 with total page 656 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a self contained, thorough introduction to the analytic and probabilistic methods of number theory. The prerequisites being reduced to classical contents of undergraduate courses, it offers to students and young researchers a systematic and consistent account on the subject. It is also a convenient tool for professional mathematicians, who may use it for basic references concerning many fundamental topics. Deliberately placing the methods before the results, the book will be of use beyond the particular material addressed directly. Each chapter is complemented with bibliographic notes, useful for descriptions of alternative viewpoints, and detailed exercises, often leading to research problems. This third edition of a text that has become classical offers a renewed and considerably enhanced content, being expanded by more than 50 percent. Important new developments are included, along with original points of view on many essential branches of arithmetic and an accurate perspective on up-to-date bibliography. The author has made important contributions to number theory and his mastery of the material is reflected in the exposition, which is lucid, elegant, and accurate. —Mathematical Reviews
Download or read book Number Theory Diophantine Problems Uniform Distribution and Applications written by Christian Elsholtz and published by Springer. This book was released on 2017-05-26 with total page 447 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is dedicated to Robert F. Tichy on the occasion of his 60th birthday. Presenting 22 research and survey papers written by leading experts in their respective fields, it focuses on areas that align with Tichy’s research interests and which he significantly shaped, including Diophantine problems, asymptotic counting, uniform distribution and discrepancy of sequences (in theory and application), dynamical systems, prime numbers, and actuarial mathematics. Offering valuable insights into recent developments in these areas, the book will be of interest to researchers and graduate students engaged in number theory and its applications.
Download or read book Multiple Dirichlet Series Automorphic Forms and Analytic Number Theory written by Solomon Friedberg and published by American Mathematical Soc.. This book was released on 2006 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multiple Dirichlet series are Dirichlet series in several complex variables. A multiple Dirichlet series is said to be perfect if it satisfies a finite group of functional equations and has meromorphic continuation everywhere. The earliest examples came from Mellin transforms of metaplectic Eisenstein series and have been intensively studied over the last twenty years. More recently, many other examples have been discovered and it appears that all the classical theorems on moments of $L$-functions as well as the conjectures (such as those predicted by random matrix theory) can now be obtained via the theory of multiple Dirichlet series. Furthermore, new results, not obtainable by other methods, are just coming to light. This volume offers an account of some of the major research to date and the opportunities for the future. It includes an exposition of the main results in the theory of multiple Dirichlet series, and papers on moments of zeta- and $L$-functions, on new examples of multiple Dirichlet
Download or read book Topics in Mathematical Analysis and Applications written by Themistocles M. Rassias and published by Springer. This book was released on 2014-10-13 with total page 811 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents significant advances in a number of theories and problems of Mathematical Analysis and its applications in disciplines such as Analytic Inequalities, Operator Theory, Functional Analysis, Approximation Theory, Functional Equations, Differential Equations, Wavelets, Discrete Mathematics and Mechanics. The contributions focus on recent developments and are written by eminent scientists from the international mathematical community. Special emphasis is given to new results that have been obtained in the above mentioned disciplines in which Nonlinear Analysis plays a central role. Some review papers published in this volume will be particularly useful for a broader readership in Mathematical Analysis, as well as for graduate students. An attempt is given to present all subjects in this volume in a unified and self-contained manner, to be particularly useful to the mathematical community.
Download or read book Arithmetic Tales written by Olivier Bordellès and published by Springer Science & Business Media. This book was released on 2012-05-31 with total page 569 pages. Available in PDF, EPUB and Kindle. Book excerpt: Number theory was once famously labeled the queen of mathematics by Gauss. The multiplicative structure of the integers in particular deals with many fascinating problems some of which are easy to understand but very difficult to solve. In the past, a variety of very different techniques has been applied to further its understanding. Classical methods in analytic theory such as Mertens’ theorem and Chebyshev’s inequalities and the celebrated Prime Number Theorem give estimates for the distribution of prime numbers. Later on, multiplicative structure of integers leads to multiplicative arithmetical functions for which there are many important examples in number theory. Their theory involves the Dirichlet convolution product which arises with the inclusion of several summation techniques and a survey of classical results such as Hall and Tenenbaum’s theorem and the Möbius Inversion Formula. Another topic is the counting integer points close to smooth curves and its relation to the distribution of squarefree numbers, which is rarely covered in existing texts. Final chapters focus on exponential sums and algebraic number fields. A number of exercises at varying levels are also included. Topics in Multiplicative Number Theory introduces offers a comprehensive introduction into these topics with an emphasis on analytic number theory. Since it requires very little technical expertise it will appeal to a wide target group including upper level undergraduates, doctoral and masters level students.
Download or read book Fourier Analysis and Convexity written by Luca Brandolini and published by Springer Science & Business Media. This book was released on 2011-04-27 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: Explores relationship between Fourier Analysis, convex geometry, and related areas; in the past, study of this relationship has led to important mathematical advances Presents new results and applications to diverse fields such as geometry, number theory, and analysis Contributors are leading experts in their respective fields Will be of interest to both pure and applied mathematicians
Download or read book Geometric Properties for Incomplete Data written by Reinhard Klette and published by Springer Science & Business Media. This book was released on 2006-03-14 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computer vision and image analysis require interdisciplinary collaboration between mathematics and engineering. This book addresses the area of high-accuracy measurements of length, curvature, motion parameters and other geometrical quantities from acquired image data. It is a common problem that these measurements are incomplete or noisy, such that considerable efforts are necessary to regularise the data, to fill in missing information, and to judge the accuracy and reliability of these results. This monograph brings together contributions from researchers in computer vision, engineering and mathematics who are working in this area. The book can be read both by specialists and graduate students in computer science, electrical engineering or mathematics who take an interest in data evaluations by approximation or interpolation, in particular data obtained in an image analysis context.
