Sur une application de la formule de Selberg–Delange

F. Ben Saïd¹ ; J.-L. Nicolas²

¹ Faculté des Sciences de Monastir Avenue de l'environnement 5000, Monastir, Tunisie
² Institut Girard Desargues, UMR 5028 Bât. Doyen Jean Braconnier Université Claude Bernard (Lyon 1) 21 Avenue Claude Bernard F-69622 Villeurbanne Cedex, France

Colloquium Mathematicum, Tome 98 (2003) no. 2, pp. 223-247

Résumé

E. Landau has given an asymptotic estimate for the number of integers up to $x$ whose prime factors all belong to some arithmetic progressions. In this paper, by using the Selberg–Delange formula, we evaluate the number of elements of somewhat more complicated sets. For instance, if $\omega(m)$ (resp. ${\mit\Omega}(m)$) denotes the number of prime factors of $m$ without multiplicity (resp. with multiplicity), we give an asymptotic estimate as $x\to \infty$ of the number of integers $m$ satisfying $2^{\omega(m)}m\le x$, all prime factors of $m$ are congruent to $3$, $5$ or $6$ modulo $7$, ${\mit\Omega}(m)\equiv i \pmod{2}$ (where $i=0$ or $1$), and $m\equiv l \pmod{b}$.The above quantity has appeared in the paper \cite{BNSL} to estimate the number of elements up to $x$ of the set $\cal A$ of positive integers containing $1$, $2$ and $3$ and such that the number $p({\cal A},n)$ of partitions of $n$ with parts in $\cal A$ is even, for all $n\ge 4$.

Détail

DOI : 10.4064/cm98-2-8

Mots-clés : landau has given asymptotic estimate number integers whose prime factors belong arithmetic progressions paper using selberg delange formula evaluate number elements somewhat complicated sets instance omega resp mit omega denotes number prime factors without multiplicity resp multiplicity asymptotic estimate infty number integers satisfying omega prime factors congruent modulo mit omega equiv pmod where equiv pmod above quantity has appeared paper cite bnsl estimate number elements set cal positive integers containing number cal partitions parts cal even