## The Prime Numbers and Their Distribution (Google eBook)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 Riemann zeta function and its connections to number theory. In the first chapter, the familiar link between $\zeta(s)$ and the distribution of primes is covered with remarkable efficiency and intuition. The second 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 thechapter 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 from the French edition. |

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### Contents

Genesis From Euclid to Chebyshev | 1 |

The Riemann Zeta Function | 29 |

Stochastic Distribution of Prime Numbers | 51 |

An Elementary Proof of the Prime Number Theorem | 77 |

The Major Conjectures | 105 |

### Common terms and phrases

2eros of C(s Abel summation ahnost analytic continuation analytic function argument arithmetic functions arithmetic progressions asymptotic formula behaviour Brun's calculation Chapter Chebyshev complex convolution coprime Cramer's model deduce defined denotes Dickman's function Dirichlet series distribution of primes elementary proof equivalent error term established estimate Euler's formula exceeding existence factori2ation finite follows forumla functional equation Gauss Hadamard half-plane hence heuristic implies inequality inverse ir(x Legendre Lemma li(x limsup Littlewood logarithm logn logp logx lower bound mathematics Mertens method Mobius function mod q modp modulo q multiplicative Nm(x number of integers number of prime obtain polynomial prime factors prime number theorem provides Quadratic residues random real number result Riemann hypothesis satisfies sequence showed si2e sieve of Eratosthenes sufficient suitable constant summatory function tend to infinity tion trivial TT(x twin prime conjecture uniform distribution uniformly distributed modulo upper bound Vallee-Poussin values variable write yields Z/mZ Z/pZ