The number of positive integers ≤x and free of prime factors >y
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The number defined by the title is denoted by Ψ(x, y). Let u = log x log y and let ρ{variant}(u) be the function determined by ρ{variant}(u) = 1, 0 ≤ u ≤ 1, uρ{variant}′(u) =  ρ{variant}(u  1), u > 1. We prove the following:. Theorem. For x sufficiently large and log y ≥ (log log x)2, Ψ(x,y) ≫ xρ{variant}(u) while for 1 + log log x ≤ log y ≤ (log log x)2, and ε > 0, Ψ(x, y) ≫ ε xρ{variant}(u) exp(u exp((log y)( 3 5  ε))). The proof uses a weighted lower approximation to Ψ(x, y), a reinterpretation of this sum in probability terminology, and ultimately largedeviation methods plus the BerryEsseen theorem. © 1985.
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