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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.
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