The Development Of Prime Number Theory
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The Development Of Prime Number Theory
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Author : Wladyslaw Narkiewicz
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14
The Development Of Prime Number Theory written by Wladyslaw Narkiewicz and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-14 with Mathematics categories.
1. People were already interested in prime numbers in ancient times, and the first result concerning the distribution of primes appears in Euclid's Elemen ta, where we find a proof of their infinitude, now regarded as canonical. One feels that Euclid's argument has its place in The Book, often quoted by the late Paul ErdOs, where the ultimate forms of mathematical arguments are preserved. Proofs of most other results on prime number distribution seem to be still far away from their optimal form and the aim of this book is to present the development of methods with which such problems were attacked in the course of time. This is not a historical book since we refrain from giving biographical details of the people who have played a role in this development and we do not discuss the questions concerning why each particular person became in terested in primes, because, usually, exact answers to them are impossible to obtain. Our idea is to present the development of the theory of the distribu tion of prime numbers in the period starting in antiquity and concluding at the end of the first decade of the 20th century. We shall also present some later developments, mostly in short comments, although the reader will find certain exceptions to that rule. The period of the last 80 years was full of new ideas (we mention only the applications of trigonometrical sums or the advent of various sieve methods) and certainly demands a separate book.
The Prime Number Theorem
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Author : G. J. O. Jameson
language : en
Publisher: Cambridge University Press
Release Date : 2003-04-17
The Prime Number Theorem written by G. J. O. Jameson and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-04-17 with Mathematics categories.
At first glance the prime numbers appear to be distributed in a very irregular way amongst the integers, but it is possible to produce a simple formula that tells us (in an approximate but well defined sense) how many primes we can expect to find that are less than any integer we might choose. The prime number theorem tells us what this formula is and it is indisputably one of the great classical theorems of mathematics. This textbook gives an introduction to the prime number theorem suitable for advanced undergraduates and beginning graduate students. The author's aim is to show the reader how the tools of analysis can be used in number theory to attack a 'real' problem, and it is based on his own experiences of teaching this material.
Number Theory And Its History
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Author : Oystein Ore
language : en
Publisher: Courier Corporation
Release Date : 2012-07-06
Number Theory And Its History written by Oystein Ore and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-07-06 with Mathematics categories.
Unusually clear, accessible introduction covers counting, properties of numbers, prime numbers, Aliquot parts, Diophantine problems, congruences, much more. Bibliography.
The Distribution Of Prime Numbers
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Author : Albert Edward Ingham
language : en
Publisher: Cambridge University Press
Release Date : 1990-09-28
The Distribution Of Prime Numbers written by Albert Edward Ingham and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990-09-28 with Mathematics categories.
Originally published in 1934, this volume presents the theory of the distribution of the prime numbers in the series of natural numbers. Despite being long out of print, it remains unsurpassed as an introduction to the field.
Additive Theory Of Prime Numbers
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Author : Luogeng Hua
language : en
Publisher: American Mathematical Soc.
Release Date : 2009-12-04
Additive Theory Of Prime Numbers written by Luogeng Hua and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-12-04 with Mathematics categories.
Loo-Keng Hua was a master mathematician, best known for his work using analytic methods in number theory. In particular, Hua is remembered for his contributions to Waring's Problem and his estimates of trigonometric sums. Additive Theory of Prime Numbers is an exposition of the classic methods as well as Hua's own techniques, many of which have now also become classic. An essential starting point is Vinogradov's mean-value theorem for trigonometric sums, which Hua usefully rephrases and improves. Hua states a generalized version of the Waring-Goldbach problem and gives asymptotic formulas for the number of solutions in Waring's Problem when the monomial $x^k$ is replaced by an arbitrary polynomial of degree $k$. The book is an excellent entry point for readers interested in additive number theory. It will also be of value to those interested in the development of the now classic methods of the subject.
Rational Number Theory In The 20th Century
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Author : Władysław Narkiewicz
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-09-02
Rational Number Theory In The 20th Century written by Władysław Narkiewicz and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-09-02 with Mathematics categories.
The last one hundred years have seen many important achievements in the classical part of number theory. After the proof of the Prime Number Theorem in 1896, a quick development of analytical tools led to the invention of various new methods, like Brun's sieve method and the circle method of Hardy, Littlewood and Ramanujan; developments in topics such as prime and additive number theory, and the solution of Fermat’s problem. Rational Number Theory in the 20th Century: From PNT to FLT offers a short survey of 20th century developments in classical number theory, documenting between the proof of the Prime Number Theorem and the proof of Fermat's Last Theorem. The focus lays upon the part of number theory that deals with properties of integers and rational numbers. Chapters are divided into five time periods, which are then further divided into subject areas. With the introduction of each new topic, developments are followed through to the present day. This book will appeal to graduate researchers and student in number theory, however the presentation of main results without technicalities will make this accessible to anyone with an interest in the area.
History Of The Theory Of Numbers Volume Ii
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Author : Leonard Eugene Dickson
language : en
Publisher: Courier Corporation
Release Date : 2005-06-07
History Of The Theory Of Numbers Volume Ii written by Leonard Eugene Dickson and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-06-07 with Mathematics categories.
The three-volume series History of the Theory of Numbers is the work of the distinguished mathematician Leonard Eugene Dickson, who taught at the University of Chicago for four decades and is celebrated for his many contributions to number theory and group theory. This second volume in the series, which is suitable for upper-level undergraduates and graduate students, is devoted to the subject of diophantine analysis. It can be read independently of the preceding volume, which explores divisibility and primality, and volume III, which examines quadratic and higher forms. Featured topics include polygonal, pyramidal, and figurate numbers; linear diophantine equations and congruences; partitions; rational right triangles; triangles, quadrilaterals, and tetrahedra; the sums of two, three, four, and n squares; the number of solutions of quadratic congruences in n unknowns; Liouville's series of eighteen articles; the Pell equation; squares in arithmetical or geometrical progression; equations of degrees three, four, and n; sets of integers with equal sums of like powers; Waring's problem and related results; Fermat's last theorem; and many other related subjects. Indexes of authors cited and subjects appear at the end of the book.
Number Theory
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Author :
language : en
Publisher: Academic Press
Release Date : 1986-05-05
Number Theory written by and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1986-05-05 with Mathematics categories.
This book is written for the student in mathematics. Its goal is to give a view of the theory of numbers, of the problems with which this theory deals, and of the methods that are used. We have avoided that style which gives a systematic development of the apparatus and have used instead a freer style, in which the problems and the methods of solution are closely interwoven. We start from concrete problems in number theory. General theories arise as tools for solving these problems. As a rule, these theories are developed sufficiently far so that the reader can see for himself their strength and beauty, and so that he learns to apply them. Most of the questions that are examined in this book are connected with the theory of diophantine equations - that is, with the theory of the solutions in integers of equations in several variables. However, we also consider questions of other types; for example, we derive the theorem of Dirichlet on prime numbers in arithmetic progressions and investigate the growth of the number of solutions of congruences.
A Primer Of Analytic Number Theory
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Author : Jeffrey Stopple
language : en
Publisher: Cambridge University Press
Release Date : 2003-06-23
A Primer Of Analytic Number Theory written by Jeffrey Stopple and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-06-23 with Mathematics categories.
This 2003 undergraduate introduction to analytic number theory develops analytic skills in the course of studying ancient questions on polygonal numbers, perfect numbers and amicable pairs. The question of how the primes are distributed amongst all the integers is central in analytic number theory. This distribution is determined by the Riemann zeta function, and Riemann's work shows how it is connected to the zeroes of his function, and the significance of the Riemann Hypothesis. Starting from a traditional calculus course and assuming no complex analysis, the author develops the basic ideas of elementary number theory. The text is supplemented by series of exercises to further develop the concepts, and includes brief sketches of more advanced ideas, to present contemporary research problems at a level suitable for undergraduates. In addition to proofs, both rigorous and heuristic, the book includes extensive graphics and tables to make analytic concepts as concrete as possible.
Prime Numbers And The Riemann Hypothesis
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Author : Barry Mazur
language : en
Publisher: Cambridge University Press
Release Date : 2016-04-11
Prime Numbers And The Riemann Hypothesis written by Barry Mazur and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-04-11 with Mathematics categories.
This book introduces prime numbers and explains the famous unsolved Riemann hypothesis.