Gérald Tenenbaum and Michel Mendès France
Preface to the English Edition ixPreface to the French Edition xiNotation and conventions xviiChapter 1. Genesis: From Euclid to Chebyshev 10. Introduction 11. Canonical decomposition 42. Congruences 53. Cryptographic intermezzo: public key systems 84. Quadratic residues 115. Return to the infinitude of the set of primes 126. The sieve of Eratosthenes 147. The Chebyshev theorems 168. Mertens’ theorems 219. Brun’s sieve and the twin prime conjecture 25Chapter 2. The Riemann Zeta Function 290. Introduction 29viiviii Contents1. Euler’s product 302. Analytic continuation 323. The line s = 1 and the prime number theorem 384. The Riemann hypothesis 425. Arithmetic consequences of information on the zeros 46Chapter 3. Stochastic Distribution of Prime Numbers 510. Introduction 511. Arithmetic progressions 522. Cram´er’s model 613. Uniform distribution modulo one 674. Geometric vision 72Chapter 4. An Elementary Proof of the Prime Number Theorem 770. Introduction 771. Integration by parts 802. Convolution of arithmetic functions 813. The M¨obius function 854. The mean value of the M¨obius function and the prime number theorem 885. Integers free of large, or small, prime factors 926. Dickman’s function 967. Daboussi’s proof, revisited 99Chapter 5. The Major Conjectures 105Further reading 113