By Harold G. Diamond

As chance and combinatorics have penetrated the material of mathematical task, sieve equipment became extra flexible and complex and lately have performed an element in probably the most astonishing mathematical discoveries. approximately 100 years have handed given that Viggo Brun invented his well-known sieve, and using sieve tools is continually evolving. Many arithmetical investigations come upon a combinatorial challenge that calls for a sieving argument, and this tract bargains a latest and trustworthy advisor in such occasions. the speculation of upper dimensional sieves is carefully explored, and examples are supplied all through. A Mathematica® software program package deal for sieve-theoretical calculations is equipped at the authors' site. To extra profit readers, the Appendix describes tools for computing sieve features.

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**Example text**

3 Prime (/-tuples Before we begin our account in earnest, we consider another example more relevant to our main objective. The inspiration for this example is the famous twin prime conjecture, which asserts that there are infinitely many pairs of positive integers (n, n+2), which are both prime numbers. The sieve method of Brun broke new ground by producing an upper bound for the number of pairs of twin primes in any interval [1, x], but the original conjecture remains unproved. 3 Prime g-tuples 7 Except for the example (2, 3), there is no other pair of primes of the form (n, n + a) for a an odd number, since one member of the pair is then even.

40 The Fundamental Lemma We have \Ad\ •= \{n: 1 < n < x, H(n) = 0 mod d}\ = p{d) (x/d + 9), \6\ < 1, so that X = x, co(d) = p(d), and rA(d) = \Ad\ - (p(d)/d)x, \rA(d)\ < By the Chinese Remainder Theorem ([HW79], Theorem 121), p is a multiplicative function. Further, by Lagrange's theorem, p(p) is at most the degree of H for every prime p; of course, by hypothesis, p(p) < p always. 1) of Landau, we have the Mertens-type relation P p

L °ZWl - — The parameter K is clearly not unique—if J7(K) holds for some number K, then it holds for any K' > K. Nevertheless, in most sieve problems the minimal K is known and we refer to it as the dimension, or sifting density, of the problem. Problems of dimension 1 are especially important and we refer to them as linear. 2 with K = g. By adjusting the bound A if necessary, we may assume that w\ > g + 1. 6) E ^-^ — l o g ^ + 0 ( ^ - 1 w\