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Sur une application de la formule de Selberg-Delange

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Ben Saïd F., Nicolas J.-L.

Colloquium Mathematicum

2003

tom 98

nr 2

223-247

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 $2^{ω(m)}m ≤ x$, all prime factors of m are congruent to 3, 5 or 6 modulo 7, Ω(m) ≡ i (mod 2)$ (where i = 0 or 1), and m ≡ l (mod b). The above quantity has appeared in the paper [3] to estimate the number of elements up to x of the set 𝓐 of positive integers containing 1, 2 and 3 and such that the number p(𝓐,n) of partitions of n with parts in 𝓐 is even, for all n ≥ 4.

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1 Colloquium Mathematicum

1 Ben Saïd F.

1 Nicolas J.-L.

1 2003

Wyniki wyszukiwania

Sur une application de la formule de Selberg-Delange