By Harold G. Diamond

ISBN-10: 0511438656

ISBN-13: 9780511438653

ISBN-10: 0521894875

ISBN-13: 9780521894876

As chance and combinatorics have penetrated the cloth of mathematical task, sieve tools became extra flexible and complex and in recent times have performed a component in one of the most outstanding mathematical discoveries. approximately 100 years have handed for the reason that Viggo Brun invented his recognized sieve, and using sieve tools is consistently evolving. Many arithmetical investigations stumble upon a combinatorial challenge that calls for a sieving argument, and this tract bargains a latest and trustworthy consultant in such events. the idea of upper dimensional sieves is punctiliously explored, and examples are supplied all through. A Mathematica® software program package deal for sieve-theoretical calculations is equipped at the authors' web site. To additional gain readers, the Appendix describes tools for computing sieve services.

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Hg(n) be distinct irreducible polynomials with integer coefficients and write H(n) = h\{n) • • • hg(n). Let p(d) denote the number of solutions of the congruence H(n) = 0 mod d that are incongruent mod d, and assume that p(p) < p for all primes p. , qg are all quasi-primes relative to x and u. , hg. Proof. Let A — {H(n) : 1 < n < x} and take V to be the set of all primes. We may assume that all the polynomials hi have positive degree; for if any one had zero degree, then the condition that p(p) < p for all p would imply that the polynomial was identically 1, and the expression on the left in the statement of the example would then be 0 and the result trivial.

6), but Iwaniec was the first to make effective use of it. 1. 1; otherwise we do not discuss Brun's method, but instead refer the reader to the Brun-Hooley method [Hoo94, FHOO], a simple but effective extension of it. 1 as Richert's Fundamental Identity and derives from it, as does Harman [Hrm96], diophantine inequalities for primes that improve on results given by the better-known Vaughan identity. 1 joins a group of simple identities that have played an important part in modern prime number theory.

4 The £l(n) condition We introduce at this point a weak average condition on OJ(-) that is to hold throughout. 3. We say that a sieve problem satisfies the J7(K;) condition provided there exist constants K, > 1, A > 1 such that W1 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.

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A Higher-Dimensional Sieve Method: With Procedures for Computing Sieve Functions by Harold G. Diamond

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