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The Prime Numbers and Their Distribution
Gérald Tenenbaum, Université Henri Poincaré, Nancy I, France, and Michel Mendès France, Université Bordeaux I, France
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Student Mathematical Library
2000; 115 pp; softcover
Volume: 6
ISBN-10: 0-8218-1647-0
ISBN-13: 978-0-8218-1647-9
List Price: US$20 Member Price: US$16
Order Code: STML/6

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 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.

Reviews

"The authors have succeeded in writing an interesting volume that can be recommended to students ... describe various aspects of prime number theory from the point of view of randomness, giving to the book a specific charm."

"A wealth of information ... The treatment is concise and the level is high. The authors have chosen to highlight some of the most important points of the area, and the exposition and the translation are excellent. Reading this book is equivalent to ascending a major summit."

-- MAA Monthly

"This is a very attractive introduction to prime number theory ... presentation is clear and concise ... [includes] material which has not previously appeared in a book. The proof [in Chapter 4] is an astonishing display of recent techniques in analytic number theory ...

"Wonderfully written, and the authors have the confidence to frequently express their delight with the subject and the sheer fun of exploring the philosophical ideas that underlie the investigation of prime numbers."

-- Mathematical Reviews

"Nicely written ... It is a pleasure to read this booklet, written by experts of number theory. Due to the many results, the elegant proofs, and the informal explanations of ideas, it is highly recommended to study this small monograph thoroughly."

-- Zentralblatt MATH

From reviews of the French edition ...

"This is a short introductory book on analytic number theory. The prerequisites are quite modest, but it still contains an impressive amount of information. A multitude of results is included, some of which were proved just recently ... this book is very well written. It is fun to read and at the same time presents most of the fundamental concepts and ideas in analytic number theory."

-- Mathematical Reviews

"The reviewer recommends it to all interested readers."

-- Zentralblatt MATH

"A wonderful book ... sweeping in scope, and ambitious ... a fearsome panorama of topics is attacked ... a thoroughly modern book, in the best sense of the phrase: it brings a beautiful collection of results in analytic number theory together ... some marvellous avant garde stuff."

-- MAA Online