ON VALUES TAKEN BY THE LARGEST PRIME FACTOR OF SHIFTED PRIMES (II)
Résumé
Denote by P the set of all primes and by P (n) the largest prime factor of integer n 1 with the convention P (1) = 1. Let η 0 ≈ 2.1426 be the unique positive zero of the equation η − 1 − 4η log(η − 1) = 0 in (1, ∞). Very recently Wu proved that for η ∈ (32 17 , η 0) there is a constant c(η) > 1 such that for each fixed non-zero integer a ∈ Z * the set (0.1) P a,c,η := {p ∈ P : p = P (q − a) for some prime q with p η < q c(η)p η } has relative density 1 in P. In this short note, we shall further extend the domain of η at the cost of obtaining a lower bound in place of an asymptotic formula, by showing that for each η ∈ [η 0 , 1 + 4 √ e) the set P a,c,η has relative positive density in P.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)
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