Sur la répartition des valeurs des fonctions arithmétiques. Le nombre de facteurs premiers d'un entier

✍ Scribed by Hsien-Kuei Hwang

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 265 KB
• Volume: 69
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1997.2216

✦ Synopsis

This paper is concerned with the quantity N(x, m), the number of positive integers n, 1 n x, for which 0(n)=m, where 0(n) denotes the total number of prime factors (counted with multiplicities) of n. The main purpose of this article is to present three powerful analytic methods, due, respectively, to Selberg, van der Waerden, and the author, the combination of which allows one to completely characterize the asymptotic behavior of the quantity N(x, m) as the second parameter varies through all its possible values, namely 1 m (log x)Âlog 2. These methods constitute, in a certain sense, a compact and effective set of analytical tools and apply also to the distribution function associated with n(x, m). All these methods are quite general and applicable to many other arithmetic functions.

1998 Academic Press j 2, ouÁ les p j sont premiers, p 1 <p 2 < } } } <p k , et les

Conside rons la quantite N(x, m) := :

A partir de cette de finition, on consideÁ re la suite de variables ale atoires ! n , qui prennent les valeurs 0(k), 1 k n, avec probabilite n &1 .