The number of positive integers x and free of prime factors >y
Academic Article
Overview
Identity
Additional Document Info
Other
View All
Overview
abstract
The number defined by the title is denoted by (x, y). Let u = log x log y and let {variant}(u) be the function determined by {variant}(u) = 1, 0 u 1, u{variant}(u) = - {variant}(u - 1), u > 1. We prove the following:. Theorem. For x sufficiently large and log y (log log x)2, (x,y) x{variant}(u) while for 1 + log log x log y (log log x)2, and > 0, (x, y) x{variant}(u) exp(-u exp(-(log y)( 3 5 - ))). The proof uses a weighted lower approximation to (x, y), a reinterpretation of this sum in probability terminology, and ultimately large-deviation methods plus the Berry-Esseen theorem. 1985.