The Prime Number Theorem
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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.
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 (in an approximate but well-defined sense) how many primes can be found that are less than any integer. The prime number theorem tells what this formula is and it is indisputably one of the the great classical theorems of mathematics. This textbook introduces the prime number theorem and is suitable for advanced undergraduates and beginning graduate students. The author deftly shows how analytical tools can be used in number theory to attack a 'real' problem.
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.
The Development Of Prime Number Theory
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Author : Wladyslaw Narkiewicz
language : en
Publisher: Springer Science & Business Media
Release Date : 2000-04-14
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 2000-04-14 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 Numbers And Their Distribution
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Author : Gerald Tenenbaum
language : en
Publisher: American Mathematical Soc.
Release Date : 2000
The Prime Numbers And Their Distribution written by Gerald Tenenbaum 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 2000 with Mathematics categories.
One notable new direction this century in the study of primes has been the influx of ideas from probability. The goal of this book is to provide insights into the prime numbers and to describe how a sequence so tautly determined can incorporate such a striking amount of randomness. The book opens with some classic topics of number theory. It ends with a discussion of some of the outstanding conjectures in number theory. In between are an excellent chapter on the stochastic properties of primes and a walk through an elementary proof of the Prime Number Theorem. This book is suitable for anyone who has had a little number theory and some advanced calculus involving estimates. Its engaging style and invigorating point of view will make refreshing reading for advanced undergraduates through research mathematicians.
Number Theory
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Author : Benjamin Fine
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-06-04
Number Theory written by Benjamin Fine 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 2007-06-04 with Mathematics categories.
This book provides an introduction and overview of number theory based on the distribution and properties of primes. This unique approach provides both a firm background in the standard material as well as an overview of the whole discipline. All the essential topics are covered: fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. Analytic number theory and algebraic number theory both receive a solid introductory treatment. The book’s user-friendly style, historical context, and wide range of exercises make it ideal for self study and classroom use.
The Prime Number Theorem
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Author : Enuenwemba Obi
language : en
Publisher:
Release Date : 1958
The Prime Number Theorem written by Enuenwemba Obi and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1958 with Numbers, Prime categories.
A historical background work on prime numbers and the problems involving them takes up the first section of this thesis. The main part of the thesis deals with the problem of the distribution of primes, often formulated in terms of the number of primes less than or equal to a given number x. It is shown in elementary number theory that if the limit of [pi](x)/x/log x exists, it must have the value 1, where [pi](x) is the number of primes less than or equal to a given number x. It is also shown that for x [greater than or equal to] 2, c [pi](x)/x/log x k, where c and k are positive constants. Neither of these results implies the other, but together, they show that (1) 0 lim inf [as x- infinity] [pi](x)/x/log x [less than or equal to] 1 [less than or equal to] lim sup [as x- infinity] [pi](x)/x/log x [infinity]. This gives rise to the prime number theorem which is stated as follows: The limit of [pi](x)/x/log x =1 as x tends to infinity. For a long time only analytic proofs of this theorem were available, but in 1948, after over one hundred years of effort on the part of mathematicians, A. Selberg demonstrated an elementary proof based on his formula: (2) [Theta](x) log x + [sigma, for p less than or equal to x] log p [theta](x/p) = 2x log x = O(x), where [Theta](x) = [sigma, for p less than or equal to x) log p and O(x) is the notation which expresses the fact that [absolute value of] (f(x)/g(x) k, a constant, as x- [infinity]. First, Selgerg showed that (3) lim inf (as x- [infinity]) [pi](x)/x/log x + lim sup (as x-[infinity]) [pi](x)/x/log x = 2. Next he showed that (4) lim inf (as x-> [infinity] [pi](x)/x/log x = lim sup (as x->[infinity]) [pi](x)/x/log x. From (3) and (4), (5) lim [pi](x)/x/log x = 1, which is the prime number theorem. Next Shapiro proves the theorem by considering intervals over which the theorem is approximately true. He employs his fundamental inequality: (6) [absolute value of] (R(x)) log2 x [less than or equal to] 2 [sigma] (for n [less than or equal to x) log n [absolute value of] (R(x/n)) + O(x log x), where R(x) is defined by [psi](x) = x + R(x) and [psi](x) = [sigma] (for p[superscript alpha] [less than or equal to] x) log p. By this method, it is first shown that (7) [absolute value of] (R(x)/x) [less than or equal to] [delta], where [delta] is a small quantity. By an application of the two lemmata below, the following recursive formula for [absolute value of] (R(x)/x) is obtained: (8) [absolute value of] (R(x)/x) [less than or equal to] [delta][subscript i], where [delta][subscript i-1] - [alpha] [delta]3[subscript i-1]. Finally, [delta][subscript i] is shown to go to zero as i -> [infinity], and from this result, the prime number theorem is deduced. Lemma 1. There exists a constant, L, {an absolute constant which can be chosen so that L> 4(log2 2)), such that given any k> 1, there exists an x0 = x0(k) such that for any x [greater than or equal to] x0, we can find an integer y, (x [less than or equal to] y [less than or equal to] kx), for which [absolute value of] (R(y)/y
The Prime Number Theorem
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Author : Graham James Oscar Jameson
language : en
Publisher:
Release Date : 2003
The Prime Number Theorem written by Graham James Oscar Jameson and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with Numbers, Prime categories.
The Great Prime Number Race
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Author : Roger Plymen
language : en
Publisher: American Mathematical Soc.
Release Date : 2020-08-13
The Great Prime Number Race written by Roger Plymen 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 2020-08-13 with Education categories.
Have you ever wondered about the explicit formulas in analytic number theory? This short book provides a streamlined and rigorous approach to the explicit formulas of Riemann and von Mangoldt. The race between the prime counting function and the logarithmic integral forms a motivating thread through the narrative, which emphasizes the interplay between the oscillatory terms in the Riemann formula and the Skewes number, the least number for which the prime number theorem undercounts the number of primes. Throughout the book, there are scholarly references to the pioneering work of Euler. The book includes a proof of the prime number theorem and outlines a proof of Littlewood's oscillation theorem before finishing with the current best numerical upper bounds on the Skewes number. This book is a unique text that provides all the mathematical background for understanding the Skewes number. Many exercises are included, with hints for solutions. This book is suitable for anyone with a first course in complex analysis. Its engaging style and invigorating point of view will make refreshing reading for advanced undergraduates through research mathematicians.
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.