By Harold G. Diamond

ISBN-10: 0511438656

ISBN-13: 9780511438653

ISBN-10: 0521894875

ISBN-13: 9780521894876

As likelihood and combinatorics have penetrated the cloth of mathematical task, sieve equipment became extra flexible and complex and lately have performed an element in essentially the most surprising mathematical discoveries. approximately 100 years have handed considering that Viggo Brun invented his recognized sieve, and using sieve tools is continually evolving. Many arithmetical investigations stumble upon a combinatorial challenge that calls for a sieving argument, and this tract deals a latest and trustworthy advisor in such events. 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 additional profit readers, the Appendix describes equipment for computing sieve services.

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Extra info for A Higher-Dimensional Sieve Method: With Procedures for Computing Sieve Functions

Example text

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\2.

1 (with (7 = 1 in the statement of the theorem) and A — {n: 1 < n < X}, we have as X ->• 00. 1; we obtain in fact a good quality "quasi-prime number theorem," which shows that the quasi-primes up to X are hardly more dense than the primes. 1 (again with q — 1) to quasi-primes. 7. , 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.

1) and derive a very useful expression for the difference d\(n,P) d\(n,P) that helps to suggest good choices for x- Given an integer d > 1, we write p(d) for the smallest prime factor of d and p+(d) for the largest prime factor. Also, we set p(l) := oo and p + ( l ) = 1. 1) X(1):=O, Then x an (d > 1). W) ••= X(d/p(d)) - X(d) d X a r e connected by the following relation. 1. 1). 4) £><,d) = d\n / , lJ'(d)x(d) d\n + 2_^ MQ d\n p(d)—p(n) 19 p\n p

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

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