“The” Hardy-Littlewood Circle Method and Applications
Author: Paula Schiesser
Publisher:
Published: 2021
Total Pages: 0
ISBN-13:
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Cubic Forms and the Circle Method
Author: Tim Browning
Publisher: Springer Nature
Published: 2021-11-19
Total Pages: 175
ISBN-13: 3030868729
DOWNLOAD EBOOKThe Hardy–Littlewood circle method was invented over a century ago to study integer solutions to special Diophantine equations, but it has since proven to be one of the most successful all-purpose tools available to number theorists. Not only is it capable of handling remarkably general systems of polynomial equations defined over arbitrary global fields, but it can also shed light on the space of rational curves that lie on algebraic varieties. This book, in which the arithmetic of cubic polynomials takes centre stage, is aimed at bringing beginning graduate students into contact with some of the many facets of the circle method, both classical and modern. This monograph is the winner of the 2021 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.
An Introduction to the Circle Method
Author: M. Ram Murty
Publisher: American Mathematical Society
Published: 2023-06-15
Total Pages: 280
ISBN-13: 1470472031
DOWNLOAD EBOOKThe circle method, pioneered by Ramanujan and Hardy in the early 20th century, has over the past 100 years become part of the standard tool chest of analytic number theory. Its scope of applications is ever-expanding, and the subject continues to see important breakthroughs. This book provides an introduction to the circle method that is accessible to undergraduate students with no background in number theory. The authors' goal is to show the students the elegance of the circle method and at the same time give a complete solution of the famous Waring problem as an illustration of the method. The first half of this book is a curated introduction to elementary number theory with an emphasis on topics needed for the second half. The second half showcases the two most “classic” applications of the circle method, to Waring's problem (following Hardy–Littlewood–Hua) and to Goldbach's conjectures (following Vinogradov, with improvements by Vaughan). This text is suitable for a one-semester undergraduate course or for independent study and will be a great entry point into this fascinating area of research.
The Hardy-Littlewood Method
Author:
Publisher: Cambridge University Press
Published:
Total Pages: 248
ISBN-13: 0521573475
DOWNLOAD EBOOKApplications of the Hardy Littlewood Method
Author: Robert C. Vaughan
Publisher:
Published: 1998
Total Pages:
ISBN-13:
DOWNLOAD EBOOKTauberian Theory
Author: Jacob Korevaar
Publisher: Springer Science & Business Media
Published: 2013-03-09
Total Pages: 497
ISBN-13: 3662102250
DOWNLOAD EBOOKTauberian theory compares summability methods for series and integrals, helps to decide when there is convergence, and provides asymptotic and remainder estimates. The author shows the development of the theory from the beginning and his expert commentary evokes the excitement surrounding the early results. He shows the fascination of the difficult Hardy-Littlewood theorems and of an unexpected simple proof, and extolls Wiener's breakthrough based on Fourier theory. There are the spectacular "high-indices" theorems and Karamata's "regular variation", which permeates probability theory. The author presents Gelfand's elegant algebraic treatment of Wiener theory and his own distributional approach. There is also a new unified theory for Borel and "circle" methods. The text describes many Tauberian ways to the prime number theorem. A large bibliography and a substantial index round out the book.
Quantitative Arithmetic of Projective Varieties
Author: Timothy D. Browning
Publisher: Springer Science & Business Media
Published: 2009-12-21
Total Pages: 168
ISBN-13: 3034601298
DOWNLOAD EBOOKThis book examines the range of available tools from analytic number theory that can be applied to study the density of rational points on projective varieties.
Harmonic Analysis on Symmetric Spaces and Applications I
Author: Audrey Terras
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 353
ISBN-13: 1461251281
DOWNLOAD EBOOKSince its beginnings with Fourier (and as far back as the Babylonian astron omers), harmonic analysis has been developed with the goal of unraveling the mysteries of the physical world of quasars, brain tumors, and so forth, as well as the mysteries of the nonphysical, but no less concrete, world of prime numbers, diophantine equations, and zeta functions. Quoting Courant and Hilbert, in the preface to the first German edition of Methods of Mathematical Physics: "Recent trends and fashions have, however, weakened the connection between mathematics and physics. " Such trends are still in evidence, harmful though they may be. My main motivation in writing these notes has been a desire to counteract this tendency towards specialization and describe appli cations of harmonic analysis in such diverse areas as number theory (which happens to be my specialty), statistics, medicine, geophysics, and quantum physics. I remember being quite surprised to learn that the subject is useful. My graduate eduation was that of the 1960s. The standard mathematics graduate course proceeded from Definition 1. 1. 1 to Corollary 14. 5. 59, with no room in between for applications, motivation, history, or references to related work. My aim has been to write a set of notes for a very different sort of course.
Diophantine Equations and Inequalities in Algebraic Number Fields
Author: Yuan Wang
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 185
ISBN-13: 3642581714
DOWNLOAD EBOOKThe circle method has its genesis in a paper of Hardy and Ramanujan (see [Hardy 1])in 1918concernedwiththepartitionfunction andtheproblemofrep resenting numbers as sums ofsquares. Later, in a series of papers beginning in 1920entitled "some problems of'partitio numerorum''', Hardy and Littlewood (see [Hardy 1]) created and developed systematically a new analytic method, the circle method in additive number theory. The most famous problems in ad ditive number theory, namely Waring's problem and Goldbach's problem, are treated in their papers. The circle method is also called the Hardy-Littlewood method. Waring's problem may be described as follows: For every integer k 2 2, there is a number s= s(k) such that every positive integer N is representable as (1) where Xi arenon-negative integers. This assertion wasfirst proved by Hilbert [1] in 1909. Using their powerful circle method, Hardy and Littlewood obtained a deeper result on Waring's problem. They established an asymptotic formula for rs(N), the number of representations of N in the form (1), namely k 1 provided that 8 2 (k - 2)2 - +5. Here
Multiplicative Number Theory I
Author: Hugh L. Montgomery
Publisher: Cambridge University Press
Published: 2007
Total Pages: 574
ISBN-13: 9780521849036
DOWNLOAD EBOOKA 2006 text based on courses taught successfully over many years at Michigan, Imperial College and Pennsylvania State.
