On integers free of large prime factors

Abstract

The number Ψ ( x , y ) \Psi (x,y) of integers ≤ x \leq x and free of prime factors > y > y has been given satisfactory estimates in the regions y ≤ ( log ⁡ x ) 3 / 4 − ε y \leq {(\log x)^{3/4 - \varepsilon }} and y > exp ⁡ { ( log ⁡ log ⁡ x ) 5 / 3 + ε } y > \exp \{ {(\log \log x)^{5/3 + \varepsilon }}\} . In the intermediate range, only very crude estimates have been obtained so far. We close this "gap" and give an expression which approximates Ψ ( x , y ) \Psi (x,y) uniformly for x ≥ y ≥ 2 x \geq y \geq 2 within a factor 1 + O ( ( log ⁡ y ) / ( log ⁡ x ) + ( log ⁡ y ) / y ) 1 + O((\log y)/(\log x) + (\log y)/y) . As an application, we derive a simple formula for Ψ ( c x , y ) / Ψ ( x , y ) \Psi (cx,y)/\Psi (x,y) , where 1 ≤ c ≤ y 1 \leq c \leq y . We also prove a short interval estimate for Ψ ( x , y ) \Psi (x,y) .