Sur une application de la formule de Selberg–Delange
Volume 98 / 2003
Abstract
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 $n\ge 4$.
