*(English)*Zbl 1002.11001

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 $\zeta \left(s\right)$ 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”.

##### MSC:

11-02 | Research monographs (number theory) |

11-01 | Textbooks (number theory) |

11Nxx | Multiplicative number theory |

11Lxx | Exponential sums; character sums |

11Mxx | Analytic theory of zeta and $L$-functions |

11P32 | Additive questions involving primes |

11N35 | Sieves |