Analytic number theory. I.
(Analytische Zahlentheorie I.)

*(German)*Zbl 0226.10003
Göttingen: Mathematisches Institut der Universität. 180 S. DM 6.00. (1963).

This Part I of the lecture notes consists of two chapters: Chapter I on multiplicative problems and Chapter II on additive problems of number theory.

Chapter I starts with estimates of the prime counting function \(\pi(x)\), using the approach of Chebyshev. Dirichlet’s theorem on primes in arithmetic progressions is proved. Dirichlet’s divisor problem is discussed and the corresponding theorem is proved with an error term \(O(n^{1/2})\) ( the stronger \(O\)-results of Voronoi, Van der Corput and Richert, as well as Hardy’s \(\Omega\)-theorem are quoted without proofs). The prime number theorem is proved (by complex integration) with an error term \(O(ne^{-c(\log n)^{1/10}})\) and the chapter ends with a proof of the functional equation of the Riemann zeta-function.

Except for a few examples, mainly from the theory of partition, Chapter II is devoted to Waring’s problem. After an elementary discussion of the representation of integers by sums of squares, the Hardy-Littlewood circle method is presented in detail (Vinogradov version; Weyl estimates on minor arcs). In the last section, the best results known concerning \(g(s)\) and \(G(s)\) (here called \(\gamma(s))\) (i.e., the number of \(s\)th powers sufficient for the additive representation of all, or of all sufficiently large natural integers, respectively) are indicated and the conjecture (sometimes called “the ideal Waring theorem”) \(g(s)=[(\tfrac 32)^s]+ 2^s-2\) is stated. While all this material is certainly classical, the presentation is masterly. No steps are skipped. Even an elementary result, such as the essential uniqueness of factorization of natural integers.

Chapter I starts with estimates of the prime counting function \(\pi(x)\), using the approach of Chebyshev. Dirichlet’s theorem on primes in arithmetic progressions is proved. Dirichlet’s divisor problem is discussed and the corresponding theorem is proved with an error term \(O(n^{1/2})\) ( the stronger \(O\)-results of Voronoi, Van der Corput and Richert, as well as Hardy’s \(\Omega\)-theorem are quoted without proofs). The prime number theorem is proved (by complex integration) with an error term \(O(ne^{-c(\log n)^{1/10}})\) and the chapter ends with a proof of the functional equation of the Riemann zeta-function.

Except for a few examples, mainly from the theory of partition, Chapter II is devoted to Waring’s problem. After an elementary discussion of the representation of integers by sums of squares, the Hardy-Littlewood circle method is presented in detail (Vinogradov version; Weyl estimates on minor arcs). In the last section, the best results known concerning \(g(s)\) and \(G(s)\) (here called \(\gamma(s))\) (i.e., the number of \(s\)th powers sufficient for the additive representation of all, or of all sufficiently large natural integers, respectively) are indicated and the conjecture (sometimes called “the ideal Waring theorem”) \(g(s)=[(\tfrac 32)^s]+ 2^s-2\) is stated. While all this material is certainly classical, the presentation is masterly. No steps are skipped. Even an elementary result, such as the essential uniqueness of factorization of natural integers.

Reviewer: Emil Grosswald

##### MSC:

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

11N05 | Distribution of primes |

11N13 | Primes in congruence classes |

11N37 | Asymptotic results on arithmetic functions |

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

11P05 | Waring’s problem and variants |

11P55 | Applications of the Hardy-Littlewood method |

11P81 | Elementary theory of partitions |