On values taken by the largest prime factor of shifted primes (II)

Denote by [Formula: see text] the set of all primes and by [Formula: see text] the largest prime factor of integer [Formula: see text] with the convention [Formula: see text]. Let [Formula: see text] be the unique positive solution of the equation [Formula: see text] in [Formula: see text]. Very recently Wu proved that for [Formula: see text] there is a constant [Formula: see text] such that for each fixed non-zero integer [Formula: see text] the set [Formula: see text] has relative density 1 in [Formula: see text]. In this paper, we shall further extend the domain of [Formula: see text] at the cost of obtaining a lower bound in place of an asymptotic formula, by showing that for each [Formula: see text] the set [Formula: see text] has relative positive density in [Formula: see text].

NONNEGATIVE MULTIPLICATIVE FUNCTIONS ON SIFTED SETS, AND THE SQUARE ROOTS OF −1 MODULO SHIFTED PRIMES

An oft-cited result of Peter Shiu bounds the mean value of a nonnegative multiplicative function over a coprime arithmetic progression. We prove a variant where the arithmetic progression is replaced by a sifted set. As an application, we show that the normalized square roots of −1 (mod m) are equidistributed (mod 1) as m runs through the shifted primes q − 1.