# The Prime Numbers and Their Distribution

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*Gérald Tenenbaum; Michel Mendès France*

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.

#### Readership

Advanced undergraduates, graduate students, and research mathematicians.

#### Reviews & Endorsements

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.

-- European Mathematical Society Newsletter

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

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

#### Table of Contents

# Table of Contents

## The Prime Numbers and Their Distribution

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface to the English Edition ix10 free
- Preface to the French Edition xi12 free
- Notation and conventions xvii18 free
- Chapter 1. Genesis: From Euclid to Chebyshev 122 free
- §0. Introduction 122
- §1. Canonical decomposition 425
- §2. Congruences 526
- §3. Cryptographic intermezzo: public key systems 829
- §4. Quadratic residues 1132
- §5. Return to the infinitude of the set of primes 1233
- §6. The sieve of Eratosthenes 1435
- §7. The Chebyshev theorems 1637
- §8. Mertens' theorems 2142
- §9. Brun's sieve and the twin prime conjecture 2546

- Chapter 2. The Riemann Zeta Function 2950
- Chapter 3. Stochastic Distribution of Prime Numbers 5172
- Chapter 4. An Elementary Proof of the Prime Number Theorem 7798
- §0. Introduction 7798
- §1. Integration by parts 80101
- §2. Convolution of arithmetic functions 81102
- §3. The Mobius function 85106
- §4. The mean value of the Mobius function and the prime number theorem 88109
- §5. Integers free of large, or small, prime factors 92113
- §6. Dickman's function 96117
- §7. Daboussi's proof, revisited 101122

- Chapter 5. The Major Conjectures 105126
- Further reading 113134
- Back cover Back Cover1137