SCREENING INTEGERS WITHOUT LARGE PRIME FA CTORS

Let PSI(x, y) (resp. PSI(q)(x, y)) denote the number of integars at mo st x (resp. and coprime to q) whose largest prime factor does not exce ed y. We give both optimal range of validity and remainder term for th e approximation of PSI(q)(x, y) by (phi(q)/q)PSI(x, y). This yields an extension of the range of validity of the smooth approximation of de Bruijn type given by Fouvry and the author for PSI(q)(x, y).