## A Higher-Dimensional Sieve Method: With Procedures for Computing Sieve FunctionsNearly 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 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 |

259 | |

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Ankeny—Onishi approximation argument asymptotic Buchstab’s choice coeﬁicients combinatorial computation condition contour integral convex Corollary decreasing deﬁned deﬁnition elogw error term establish evaluate Example 1.2 expression ﬁnd ﬁnite ﬁrst ﬁxed follows formula Fundamental Lemma given half integer Hence holds implies inﬁnitely integrand interval irreducible polynomials Iwaniec inner product Laplace transform last inequality left side log log log2 logp logt logw logwl logX logy logz logzo lower bound Mathematica Mertens monotonicity non-negative Notes on Chapter numerical quadrature obtain pair parameter path positive integer prime divisors prime factors prime ideals proof of Theorem Proposition 7.3 recall remainder sum restate result right side Rosser—Iwaniec saddle point satisﬁes satisfying Section Selberg Selberg sieve sieve method sieve theory sieving limit solution speciﬁc squarefree strictly increasing suﬁiciently Suppose Theorem 6.1 trivial twin prime twin prime conjecture up(u upper bound write X+(d yields zero