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 (R) 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 (R) which aid in the
computation of such functions.
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