Abstract

Let F( x,y ) (resp. F q ( x,y )) denote the number of integars at most x (resp. and coprime to q ) whose largest prime factor does not exceed y . We give both optimal range of validity and remainder term for the approximation of (resp. F q ( x,y ) by ((p( q )/ q )lF( 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 (resp. F q ( x,y ).

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