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 ω(m) (resp. Ω(m)) denotes the number of prime factors of m without multiplicity (resp. with multiplicity), we give an asymptotic estimate as x → ∞ of the number of integers m satisfying , all prime factors of m are congruent to 3, 5 or 6 modulo 7, Ω(m) ≡ i (mod 2)The above quantity has appeared in the paper [3] to estimate the number of elements up to x of the set of positive integers containing 1, 2 and 3 and such that the number p(,n) of partitions of n with parts in is even, for all n ≥ 4.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm98-2-8,
author = {F. Ben Sa\"\i d and J.-L. Nicolas},
title = {Sur une application de la formule de Selberg-Delange},
journal = {Colloquium Mathematicae},
volume = {96},
year = {2003},
pages = {223-247},
zbl = {1051.11049},
language = {fra},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm98-2-8}
}
F. Ben Saïd; J.-L. Nicolas. Sur une application de la formule de Selberg-Delange. Colloquium Mathematicae, Tome 96 (2003) pp. 223-247. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm98-2-8/