Analytic number theory. An introductory course.

*(English)*Zbl 1074.11001Monographs in Number Theory 1. River Edge, NJ: World Scientific (ISBN 981-238-938-5/hbk; 981-256-080-7/pbk). xiii, 360 p. £ 26.00; $ 42.00 pbk; £ 48.00; $ 78.00 hbk (2004).

This textbook steers a gentle course through multiplicative analytic number theory. Beginning with arithmetic functions and their summatory functions, it takes the student through the elementary proof of the prime number theorem (PNT), and then to Dirichlet series, the Wiener-Ikehara proof of the PNT, and then to the classical proof, with de la Vallée-Poussin’s error term. One then meets characters, Dirichlet $L$-functions, and the PNT for arithmetic progressions, followed by some simple oscillation theorems. The book concludes with two chapters on sieves. There are exercises scattered throughout the book, and end of chapter notes.

This book is suitable for beginning graduate students, or possibly even advanced undergraduates.

Reviewer: Roger Heath-Brown (Oxford)

##### MSC:

11-01 | Textbooks (number theory) |

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

11Nxx | Multiplicative number theory |

11N37 | Asymptotic results on arithmetic functions |

11N36 | Applications of sieve methods |

11N05 | Distribution of primes |

11M06 | $\zeta \left(s\right)$ and $L(s,\chi )$ |