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**Basic analytic number theory. Transl. from the Russian by Melvyn B. Nathanson.**
*(English)*
Zbl 0767.11001

Berlin: Springer-Verlag. xiii, 222 p. (1993).

This is a translation of the second Russian edition, published in 1983. The second edition differs from the first [Principles of analytic number theory (Moscow 1975; Zbl 0428.10019)] in many points. There is added a new Chapter I on integer points and some new proofs for various theorems.

The aim of the book is to give an introduction to important problems and methods of analytic number theory. It describes basic results, and proofs are given in detail. Not ever the best possible known estimates of the error terms in some problems are given, because the proofs are very technical and voluminous. Therefore, the book reads well and it may be used as a textbook by graduate students.

Four problems come to the fore: 1. The estimation of the number of lattice points in planar domains. 2. The distribution of primes. 3. Goldbach’s problem. 4. Waring’s problem. In connection with these problems the following important methods of analytic number theory are described: The method of complex integration, I. M. Vinogradov’s method of trigonometric functions, the circle method of G. H. Hardy and J. E. Littlewood.

Contents: I. Integer points. II. Entire functions of finite order. III. The Euler gamma function. IV. The Riemann zeta function. V. The connection between the sum of the coefficients of a Dirichlet series. VI. The method of I. M. Vinogradov in the theory of the zeta function. VII. The density of the zeros of the zeta function and the problem of the distribution of prime numbers in short intervals. VIII. Dirichlet \(L\)- functions. IX. Prime numbers in arithmetic progressions. X. The Goldbach conjecture. XI. Waring’s problem.

There are many and often difficult exercises at the end of each chapter with some hints for solutions. They should have a stimulating effect for further research. Finally, there is a table of prime numbers less than 4070 and their smallest primitive roots.

The aim of the book is to give an introduction to important problems and methods of analytic number theory. It describes basic results, and proofs are given in detail. Not ever the best possible known estimates of the error terms in some problems are given, because the proofs are very technical and voluminous. Therefore, the book reads well and it may be used as a textbook by graduate students.

Four problems come to the fore: 1. The estimation of the number of lattice points in planar domains. 2. The distribution of primes. 3. Goldbach’s problem. 4. Waring’s problem. In connection with these problems the following important methods of analytic number theory are described: The method of complex integration, I. M. Vinogradov’s method of trigonometric functions, the circle method of G. H. Hardy and J. E. Littlewood.

Contents: I. Integer points. II. Entire functions of finite order. III. The Euler gamma function. IV. The Riemann zeta function. V. The connection between the sum of the coefficients of a Dirichlet series. VI. The method of I. M. Vinogradov in the theory of the zeta function. VII. The density of the zeros of the zeta function and the problem of the distribution of prime numbers in short intervals. VIII. Dirichlet \(L\)- functions. IX. Prime numbers in arithmetic progressions. X. The Goldbach conjecture. XI. Waring’s problem.

There are many and often difficult exercises at the end of each chapter with some hints for solutions. They should have a stimulating effect for further research. Finally, there is a table of prime numbers less than 4070 and their smallest primitive roots.

Reviewer: E.KrĂ¤tzel (Jena)

### MSC:

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

11N05 | Distribution of primes |

11P05 | Waring’s problem and variants |

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

11P21 | Lattice points in specified regions |

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