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The number of unsieved integers up to x
 Acta Arith
"... Abstract. Typically, one expects that there are around x ∏ p̸∈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 mo ..."
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Abstract. Typically, one expects that there are around x ∏ p̸∈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 eγ + o(1) times the expected amount (where γ is the EulerMascheroni constant), which was improved slightly by Hildebrand [5]. Hildebrand [6] also showed that for a given value of ∏ p̸∈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 Lipschitztype bound for such counts. p≤x