The Prime Numbers and Their Distribution (Student Mathematical Library, Vol. 6) (Student Mathematical Library, V. 6) by Gerald Tenenbaum (PDF)

Description

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.

User’s Reviews

Editorial Reviews: Review “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 From the Publisher 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 f”ur Mathematik”

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I used to consider myself a mathematician; indeed at one time I expected to become a professional. I love prime numbers above all else in mathematics; the most elusive, beautiful and singular numbers ever. This is exactly what I am most interested in: their distribution, something which has fascinated mathematicians since the Greeks.No doubt this is a very deep and powerful work about the subject I care most about. Like virtually all such books on that subject, I simply cannot follow the dense rows of equations, hypotheses and lemmas which compose so much of its text. I only wish I were among the exalted who can; alas, I am not.For those who are at the top of their game, and who find number theory to be as Gauss once said “The Queen of mathematics,” this is a book for you! Unless you are a dedicated, serious student of this subject, do not attempt to tread this awesome path.

⭐”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.

⭐Dense and poorly paced and not the world’s best translationmake this an unappetizing window onto what should be abeautiful topic. Additionally, these rough notes should neverhave appeared in a ‘student’ series — only one more exampleof the problems that have plagued this series.

⭐This book is not suitable for anyone who has had a little number theory and some advanced calculus involving estimates. Complex terminology and too theorist, I miss examples and numbers and terminology should have been more explained. I didn’t like it al all

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