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**Multiplicative number theory. Revised and with a preface by Hugh L. Montgomery.
3rd ed.**
*(English)*
Zbl 1002.11001

Graduate Texts in Mathematics. 74. New York, NY: Springer. x, 177 p. DM 99.00; öS 723.00; sFr. 90.50; £34.00; $ 49.95 (2000).

This “classic” text in number theory studies the distribution of primes in arithmetic progressions, a.o. the Dirichlet theorem, estimates for zero-free regions of \(\zeta(s)\) and \(L(s,\chi)\), Siegel’s theorem, functional equations for \(L\)-functions, etc. (Chapters 1-22).

Chapters 23-29 are devoted to the Polya-Vinogradov inequality, prime number sums, exponential sums with primes, sums of three primes and culminate in treating the large sieve, Bombieri’s theorem, and an average result. For more details see the reviews of Kátai and Burgess of the first two editions (1967; Zbl 0159.06303 and 1980; Zbl 0453.10002).

The final chapter has been updated from the 2nd ed. to include more recent results by Baker, Harman, Heath-Brown, Maier and Pomerance, and a number of recent books is cited.

In conclusion, we cite from Burgess’ review: “Montgomery is to be congratulated on avoiding the introduction of any disturbing contrast in style between the original text and the section that has received total revision. The new edition is to be as highly recommended today as was the original in its time”.

Chapters 23-29 are devoted to the Polya-Vinogradov inequality, prime number sums, exponential sums with primes, sums of three primes and culminate in treating the large sieve, Bombieri’s theorem, and an average result. For more details see the reviews of Kátai and Burgess of the first two editions (1967; Zbl 0159.06303 and 1980; Zbl 0453.10002).

The final chapter has been updated from the 2nd ed. to include more recent results by Baker, Harman, Heath-Brown, Maier and Pomerance, and a number of recent books is cited.

In conclusion, we cite from Burgess’ review: “Montgomery is to be congratulated on avoiding the introduction of any disturbing contrast in style between the original text and the section that has received total revision. The new edition is to be as highly recommended today as was the original in its time”.

Reviewer: O.Ninnemann (Berlin)

### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11Nxx | Multiplicative number theory |

11Lxx | Exponential sums and character sums |

11Mxx | Zeta and \(L\)-functions: analytic theory |

11P32 | Goldbach-type theorems; other additive questions involving primes |

11N35 | Sieves |

### Keywords:

prime number theorem; Riemann zeta-function; Dirichlet \(L\)-function; cyclotomy; Vinogradov method; Gauss sums; Vaughan method; distribution of primes in arithmetic progressions; Dirichlet theorem; zero-free regions; Siegel’s theorem; functional equations; Polya-Vinogradov inequality; exponential sums with primes; sums of three primes; large sieve; Bombieri’s theorem
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\textit{H. Davenport}, Multiplicative number theory. Revised and with a preface by Hugh L. Montgomery. 3rd ed. New York, NY: Springer (2000; Zbl 1002.11001)