If we sieve the integers up to \( x \) by the primes not in some set \( P\), we expect around
\( x\prod_{p\not\in P,\ p\leq x} (1-1/p)\) integers left. It is known that one cannot get more than \(e^\gamma\) times this expectation, which we improve (to a more-or-less best possible upper bound). Hildebrand showed that over all sets \( P\) for which \( \prod_{p\not\in P,\ p\leq x} (1-1/p) >1/u \), one gets the least number of integers left unsieved if \( P \) is the set of primes bigger than some \( y\). We give a rather different proof of Hildebrand's Theorem.
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