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

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Cambridge University Press, Oct 16, 2008 - Mathematics
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Nearly a hundred years have passed since Viggo Brun invented his famous sieve, and the use of sieve methods is constantly evolving. As probability and combinatorics have penetrated the fabric of mathematical activity, sieve methods have become more versatile and sophisticated and in recent years have played a part in some of the most spectacular mathematical discoveries. Many arithmetical investigations encounter a combinatorial problem that requires a sieving argument, and this tract offers a modern and reliable guide in such situations. The theory of higher dimensional sieves is thoroughly explored, and examples are provided throughout. A Mathematica® software package for sieve-theoretical calculations is provided on the authors' website. To further benefit readers, the Appendix describes methods for computing sieve functions. These methods are generally applicable to the computation of other functions used in analytic number theory. The appendix also illustrates features of Mathematica® which aid in the computation of such functions.
 

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Contents

Section 1
3
Section 2
13
Section 3
19
Section 4
29
Section 5
41
Section 6
43
Section 7
55
Section 8
63
Section 16
135
Section 17
155
Section 18
163
Section 19
169
Section 20
184
Section 21
188
Section 22
193
Section 23
204

Section 9
67
Section 10
81
Section 11
97
Section 12
103
Section 13
116
Section 14
120
Section 15
125
Section 24
207
Section 25
217
Section 26
229
Section 27
233
Section 28
248
Section 29
259

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About the author (2008)

Harold G. Diamond is Professor Emeritus in the Department of Mathematics at the University of Illinois at Urbana-Champaign.

Heini Halberstam is Professor Emeritus in the Department of Mathematics at the University of Illinois at Urbana-Champaign.

William F. Galway's research focuses on analytic and computational number theory. He is a member of the American Mathematical Society and of the Mathematical Association of America.

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