Books : The Prime Numbers and Their Distribution (Student Mathematical Library, Vol. 6) (Student Mathematical Library, V. 6)
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Type of bind: Paperback
Dewey Decimal Number: 512.72
EAN num: 9780821816479
ISBN number: 0821816470
Label: American Mathematical Society
Manufacturer: American Mathematical Society
Quantity: 1
Page Count: 115
Printing Date: 2000-05
Publishing house: American Mathematical Society
Sale Popularity Level: 827007
Studio: American Mathematical Society
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We have been curious about numbers--and prime numbers--since antiquity. 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.
There are two ways in which the book is exceptional. First, some familiar topics are covered with refreshing insight and/or from new points of view. Second, interesting recent developments and ideas are presented that shed new light on the prime numbers and their distribution among the rest of the integers.
The book begins with a chapter covering some classic topics, such as quadratic residues and the Sieve of Eratosthenes. Also discussed are other sieves, primes in cryptography, twin primes, and more.
Two separate chapters address the asymptotic distribution of prime numbers. In the very first of these, the familiar link between $\zeta(s)$ and the distribution of primes is covered with remarkable efficiency and intuition. The later chapter presents a walk through an elementary proof of the Prime Number Theorem. To help the novice understand the 'why' of the proof, connections are made along the way with more familiar results such as Stirling's formula.
A most distinctive chapter covers the stochastic properties of prime numbers. The authors present a wonderfully clever interpretation of primes in arithmetic progressions as a phenomenon in probability. They also describe Cramér's model, which provides a probabilistic intuition for formulating conjectures that have a habit of being true. In this context, they address interesting questions about equipartition modulo $1$ for sequences involving prime numbers. The final section of the chapter compares geometric visualizations of random sequences with the visualizations for similar sequences derived from the primes. The resulting pictures are striking and illuminating. The book concludes with a chapter on the outstanding big conjectures about prime numbers.
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. This book is the English translation of the French edition.
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"The Prime Numbers and Their Distribution" presents an introduction to modern analytic number theory of prime numbers. The primary tool of the book is advanced calculus and estimation. The book was written in French and then translated to English. The purpose of the book is to understand both the deterministic and stochastic aspects of prime numbers. " ... Legendre and Gauss conjectured a harmonious distribution for the prime numbers, namely [the Prime Number Theorem (PNT)] ..." "Chapters 1, 2, and 4 are mainly devoted to regularity results while Chapter 3 essentially deals with random aspects of the distribution of prime numbers. In Chapter 5 we describe the principle conjectures [of analytic number theory of prime numbers] ... "
The major regularity results on chapter 1 includes (a) the Chebyshev theorems, (b) Merten's theorems, and (c) Bruns's sieve.
Chapter two is devoted to the Riemann Zeta function, which is closely related to the prime numbers (Euler's product) and the PNT (the Riemann Zeta function has no zero on the line sigma=1 implies the PNT). The Riemann hypothesis is also included in chapter 2.
Chapter 4 provides a modern elementary proof for the PNT. The modern proof (Daboussi's proof) utilizes the modern research tools such as (a) convolutions of arithmetic functions, (b) sieve, and (c) solution of differential-difference equations.
Chapter 3 (Stochastic Distribution of the Primes) devotes to (a) arithmetic progression (related to a Field medalist's work), (b) Cramer's model of prime numbers, (c) uniform distribution modulo one (every sufficiently large odd integer is the sum of at most three primes--the most significant step towards the Goldbach conjecture).
Chapter 5 devotes to the major conjectures of the prime numbers. They are (1) whether or not there are infinitely many prime numbers of the form n^2 + 1, (2) Goldbach's conjecture: every even number > 2 is the sum of two primes (every odd number > 5 is the sum of three primes.), (3) Chebyshev conjecture: there are more primes of the form 4m+3 than of the form 4m+1, (4) whether there are infinitely many Mersenne primes.
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This book gives a survey of some of the top results, methods, and conjectures about the distribution of prime numbers. For many results it gives complete (but very concise) proofs.
Highlights are: a sketch of Dirichlet's original proof of his theorem on the infinitude of primes in arithmetic progressions; a new (1984) elementary proof of the Prime Number Theorem due to Henri Daboussi; a brief introduction to Cramer's ideas about using probability theory to conjecture results about the distribution of primes; and a survey of current unsolved problems. Daboussi's proof is especially interesting because it introduces a number of ideas that are used over and over again in more advanced work, in particular the study of numbers free of large, or small, prime factors.
The book can be read either as a survey of what is currently known, or in more detail for a good understanding of modern methods.
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