The number of unsieved integers up to x

Abstract

Typically, one expects that there are around x p ∈P, p≤x (1 -1/p) integers up to x, all of whose prime factors come from the set P .Of course for some choices of P one may get rather more integers, and for some choices of P one may get rather less.Hall [4] showed that one never gets more than e γ + o(1) times the expected amount (where γ is the Euler-Mascheroni constant), which was improved slightly by Hildebrand [5].Hildebrand [6] also showed that for a given value of p ∈P, p≤x (1 -1/p), the smallest count that you get (asymptotically) is when P consists of all the primes up to a given point.In this paper we shall improve Hildebrand's upper bound, obtaining a result close to optimal, and also give a substantially shorter proof of Hildebrand's lower bound.As part of the proof we give an improved Lipschitz-type bound for such counts.

Locations

• Acta Arithmetica - View - PDF
• arXiv (Cornell University) - View - PDF
• DataCite API - View