Averages of certain multiplicative functions over smooth integers. (Moyennes de certaines fonctions multiplicatives sur les entiers friables.) (French) Zbl 1195.11132

Tenenbaum, Gérald; Wu, Jie

J. Reine Angew. Math. 564, 119-166 (2003).

Summary: An integer \(n\) is called smooth (without large prime factors, French: friable) when its prime number factorization consists exclusively of (relatively) small factors. Let \(P(n)\) be the largest prime factor occurring in this factorization, and let \(S(x,y)\) denote the set \(\{n\leq x, P(n)\leq y\}\). This work investigates the asymptotic behaviour of the summatory function \[ \Psi_f(x, y):=\sum_{n\in S(x,y)}f(n) \] when \(f\) is a multiplicative arithmetic function satisfying some simple and general conditions concerning its mean behaviour on primes and on powers of primes. One such condition is \(|\sum_{p\leq z}f(p)\log p-\kappa z|\leq Cz/R(z)\) \((z>1)\), where \(C\) is a constant and \(\kappa>0\); requirements on \(R\) are technical conditions too long to state here, but satisfied by any “reasonable” positive increasing function. By setting \(R(z)=(\log z)^\delta\) in the very general Theorem 2.1, the authors obtain a more general as well as more precise estimate on \(\Psi_f(x, y)\) (Corollary 2.2) than that recently obtained by J. M. Song [Acta Arith. 102, No. 2, 105–129 (2002; Zbl 0988.11041)]. Their next result (Corollary 2.3) offers a general estimate in the case where \(f(p)\) is on average very close to a constant, and contains without loss of precision estimates of the literature for particular functions f, such as the so-called Piltz functions \(\tau_k\), or the function \(\mu^2\) where \(\mu\) is the Möbius function. Then, as a further application of their first result, they establish an Erdős-Wintner theorem on smooth integers (Theorem 2.4). They finally mention an application to the case where \(f(n)\) is the characteristic function of the integers that can be represented as a sum of two squares of integers (Theorem 2.5); their estimate is uniformly valid for \(x\geq 3\), \(\exp((\log x)^{2/5+\varepsilon})\leq y\leq x\).
The paper begins with an historical introduction, and is followed by an extensive bibliography on the subject.
For Parts II–IV see (with G. Hanrot, Proc. Lond. Math. Soc. (3) 96, No. 1, 107–135 (2008; Zbl 1195.11129); Compos. Math. 144, No. 2, 339–376 (2008; Zbl 1168.11043), and CRM Proc. Lect. Notes 46, 129–141 (2008; Zbl 1182.11042).

MSC:

11N37	Asymptotic results on arithmetic functions
11N25	Distribution of integers with specified multiplicative constraints

Keywords:

integers without large prime factors; asymptotic behaviour of the summatory function; multiplicative arithmetic function

Citations:

Zbl 0988.11041; Zbl 1168.11043; Zbl 1182.11042; Zbl 1195.11129

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