Sur une application de la formule de Selberg–Delange

Volume 98 / 2003

F. Ben Saïd, J.-L. Nicolas Colloquium Mathematicum 98 (2003), 223-247 MSC: 11N25, 11N37. DOI: 10.4064/cm98-2-8


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 $n\ge 4$.


  • F. Ben SaïdFaculté des Sciences de Monastir
    Avenue de l'environnement
    5000, Monastir, Tunisie
  • J.-L. NicolasInstitut Girard Desargues, UMR 5028
    Bât. Doyen Jean Braconnier
    Université Claude Bernard (Lyon 1)
    21 Avenue Claude Bernard
    F-69622 Villeurbanne Cedex, France

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