Description
In terms of sieve theory the Brun sieve is of combinatorial type: that is, derives from a careful use of the inclusion-exclusion principle.
Let A be a set of positive integers ≤ x and let P be a set of primes. For each p in P, let Ap denote the set of elements of A divisible by p and extend this to let Ad the intersection of the Ap for p dividing d, when d is a product of distinct primes from P. Further let A1 denote A itself. Let z be a positive real number and P(z) denote the primes in P ≤ z. The object of the sieve is to estimate
We assume that |Ad| may be estimated by
where w is a multiplicative function and X = |A|. Let
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