Chapter 1.0 Some tools from real analysis 3 § 0.1 Abel summation 3 § 0.2 The Euler-Maclaurin summation formula 5 Exercises 7 Chapter 1.1 Prime numbers 9 § 1.1 Introduction 9 § 1.2 Chebyshev's estimates § 1.3 p-adic valuation of n! ' § 1.4 Mertens' first theorem § 1.5 Two new asymptotic formulae § 1.6 Mertens' formula § 1.7 Another theorem of Chebyshev Notes 20 Exercises 20 Chapter 1.2 Arithmetic functions 23 § 2.1 Definitions § 2.2 Examples § 2.3 Formal Dirichlet series § 2.4 The ring of arithmetic functions § 2.5 The Mobius inversion formulae ' § 2.6 Von Mangoldt's function ' § 2.7 Euler's totient function Notes 33 Exercises 34 Chapter 1.3 Average orders 36 § 3.1 Introduction § 3.2 Dirichlet's problem and the hyperbola method 36 § 3.3 The sum of divisors function § 3.4 Euler's totient function § 3.5 The functions LJ and fl § 3.6 Mean value of the Mobius function and the summatory functions of Chebyshev 42 § 3.7 Squarefree integers 46

L'accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 1. Estimates for ~(:r,~) 2013 a surveyWe begin by pointing out a natural but false heuristic for the size of 'A 3

The number Ψ ( x , y ) \Psi (x,y) of integers ≤ x \leq x and free of prime factors > y > y has been given satisfactory estimates in the regions y ≤ ( log ⁡ x ) 3 / 4 − ε y \leq {(\log x)^{3/4 - \varepsilon }} and y > exp ⁡ { ( log ⁡ log ⁡ x ) 5 / 3 + ε } y > \exp \{ {(\log \log x)^{5/3 + \varepsilon }}\} . In the intermediate range, only very crude estimates have been obtained so far. We close this "gap" and give an expression which approximates Ψ ( x , y ) \Psi (x,y) uniformly for x ≥ y ≥ 2 x \geq y \geq 2 within a factor 1 + O ( ( log ⁡ y ) / ( log ⁡ x ) + ( log ⁡ y ) / y ) 1 + O((\log y)/(\log x) + (\log y)/y) . As an application, we derive a simple formula for Ψ ( c x , y ) / Ψ ( x , y ) \Psi (cx,y)/\Psi (x,y) , where 1 ≤ c ≤ y 1 \leq c \leq y . We also prove a short interval estimate for Ψ ( x , y ) \Psi (x,y) .

This is a systematic account of the multiplicative structure of integers, from the probabilistic point of view. The authors are especially concerned with the distribution of the divisors, which is as fundamental and important as the additive structure of the integers, and yet until now has hardly been discussed outside of the research literature. Hardy and Ramanujan initiated this area of research and it was developed by Erdös in the thirties. His work led to some deep and basic conjectures of wide application which have now essentially been settled. This book contains detailed proofs, some of which have never appeared in print before, of those conjectures that are concerned with the propinquity of divisors. Consequently it will be essential reading for all researchers in analytic number theory.

puisque r)x ^ Q~xx = (\ogy) B . Grace a (9.5), on peut ecrire la contribution du terme principal de (2.12) sous la forme f e{r ] t)dV q {t,y)= f e( V t)\v q (t, y) + ^^] dt + f e(r,t)d{tS q (t)}.Jxly J xly *-lO &y J J xlyPar le lemme 9.2, on voit que la premiere integrale est de 1'ordre de grandeur souhaite". La seconde est egale a