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Sieve methods

Prelude multiplicative functions, Introduction to sieve methods, Interlude probabilistic number theory, The combinatorial sieve, Some applications of sieve methods, Gaps between primes, Irregularities in the distribution of primes,. As the main contents of the document "Sieve methods". Invite you to consult. | Sieve methods Dimitris Koukoulopoulos University of Montreal Fall 2012 Last update November 8 2013 si 2 Contents 0 Prelude multiplicative functions 7 0.1 Convolution of arithmetic functions. 7 0.2 Averages of multiplicative functions basic techniques. 8 0.3 Rankin s method. 11 0.4 Averages of multiplicative functions integral-delay equations. 14 1 Introduction to sieve methods 19 1.1 The sieve of Eratosthenes-Legendre. 19 1.2 General set-up. 22 1.3 Sifting dimension. 25 2 Interlude probabilistic number theory 27 2.1 The number of prime factors of an integer. 27 2.2 The Kubilius model. 29 3 The combinatorial sieve 31 3.1 Brun s pure sieve. 31 3.2 Buchstab iterations and general upper lower bound sieves. 34 3.3 The fundamental lemma of sieve methods . 37 4 Some applications of sieve methods 43 4.1 Prime values of polynomials. 44 4.2 The image of Euler s totient function. 48 4.3 The Titchmarsch-Linnik divisor problem. 50 5 Selberg s sieve 55 5.1 An optimization problem. 55 5.2 The fundamental lemma encore. 59 5.3 Applications. 62 5.4 The parity problem in sieve methods. 68 6 Smooth numbers 71 6.1 Iterative arguments and integral-delay equations. 71 6.2 Rankin s method encore. 77