Problems in analytic number theory.

*(English)*Zbl 0971.11001
Graduate Texts in Mathematics. 206. New York, NY: Springer. xvi, 452 p. (2001).

In short, the aim of this beautiful monograph on analytic number theory may best be described by the beginning of the preface of this book.

“ ‘In order to become proficient in mathematics, or in any subject,’ writes André Weil, ‘the student must realize that most topics involve only a small number of basic ideas.’ After learning these basic concepts and theorems, the student should ‘drill in routine exercises, by which the necessary reflexes in handling such concepts may be acquired. …There can be no real understanding of the basic concepts of a mathematical theory without an ability to use them intelligently and apply them to specific problems.’ Weil’s insightful observation becomes especially important at the graduate and research level. …Our goal is to acquaint the student with the methods of analytic number theory as rapidly as possible through examples and exercises.”

So, in each of the ten sections, this monograph gives important results and techniques for specific topics, together with many exercises. Then, more than 250 pages are necessary to give the solutions of these exercises. In the bibliography there are references to well-known monographs in number theory, for example those of H. Davenport [Multiplicative number theory, Revised by Hugh L. Montgomery. 3rd ed. Graduate Texts in Mathematics. 74. New York, Springer (2000; Zbl 1002.11001); 2nd ed. (1980; Zbl 0453.10002)], H. Halberstam and H.-E. Richert [Sieve Methods, London Mathematical Society Monographs. 4 (1974; Zbl 0298.10026)], J. H. Hardy and E. M. Wright [An introduction to the theory of numbers. 5th ed., Oxford (1979; Zbl 0423.10001)], N. I. Koblitz [\(p\)-adic numbers, \(p\)-adic analysis, and \(\zeta\)-functions, Graduate Texts in Mathematics 58. New York, Springer (1977; Zbl 0364.12015)], and L. C. Washington [Introduction to cyclotomic fields. 2nd ed. Graduate Texts in Mathematics. 83. New York, Springer (1997; Zbl 0966.11047)].

Quoting the chapter headings, this monograph deals with

– Arithmetic Functions (Möbius inversion, order, average order),

– Primes in Arithmetic Progressions (Abel’s summation, characters, Dirichlet’s hyperbola -- method),

– The Prime Number Theorem (Chebyshev’s results, \(\zeta(1+it) \not=0\), the Landau-Ikehara theorem -- [called the Ikehara-Wiener theorem in the book], and the prime number theorem),

– The Method of Contour Integration (which gives another proof of the prime number theorem, -- further examples for applications of the method),

– Functional Equations (Poisson’s formula, functional equation for \(\zeta(s)\) and \(L(s,\chi)\)),

– Hadamard Products (Jensen’s theorem, entire functions, zero-free regions),

– Explicit Formulas (for \(\psi(x)\), Weil’s explicit formula),

– The Selberg Class (Dirichlet series \(\sum a_n \cdot n^{-s}\) with a functional equation, with an Euler -- product, and with an estimate \(a_n ={\mathcal O}_{\varepsilon} (n^{\varepsilon})\)),

– Sieve Methods (Eratosthenes’ sieve, Brun’s sieve, Selberg’s sieve; the ‘Large Sieve’ is missing),

– \(p\)-adic Methods (Theorem of Ostrowski, Hensel’s lemma, \(p\)-adic zeta function).

Of course, it is not possible to describe adequately the wealth of material covered in this book. The reviewer thinks that every adept of number theory ought to work by himself through the problems of this monograph – certainly he will gain large benefits.

“ ‘In order to become proficient in mathematics, or in any subject,’ writes André Weil, ‘the student must realize that most topics involve only a small number of basic ideas.’ After learning these basic concepts and theorems, the student should ‘drill in routine exercises, by which the necessary reflexes in handling such concepts may be acquired. …There can be no real understanding of the basic concepts of a mathematical theory without an ability to use them intelligently and apply them to specific problems.’ Weil’s insightful observation becomes especially important at the graduate and research level. …Our goal is to acquaint the student with the methods of analytic number theory as rapidly as possible through examples and exercises.”

So, in each of the ten sections, this monograph gives important results and techniques for specific topics, together with many exercises. Then, more than 250 pages are necessary to give the solutions of these exercises. In the bibliography there are references to well-known monographs in number theory, for example those of H. Davenport [Multiplicative number theory, Revised by Hugh L. Montgomery. 3rd ed. Graduate Texts in Mathematics. 74. New York, Springer (2000; Zbl 1002.11001); 2nd ed. (1980; Zbl 0453.10002)], H. Halberstam and H.-E. Richert [Sieve Methods, London Mathematical Society Monographs. 4 (1974; Zbl 0298.10026)], J. H. Hardy and E. M. Wright [An introduction to the theory of numbers. 5th ed., Oxford (1979; Zbl 0423.10001)], N. I. Koblitz [\(p\)-adic numbers, \(p\)-adic analysis, and \(\zeta\)-functions, Graduate Texts in Mathematics 58. New York, Springer (1977; Zbl 0364.12015)], and L. C. Washington [Introduction to cyclotomic fields. 2nd ed. Graduate Texts in Mathematics. 83. New York, Springer (1997; Zbl 0966.11047)].

Quoting the chapter headings, this monograph deals with

– Arithmetic Functions (Möbius inversion, order, average order),

– Primes in Arithmetic Progressions (Abel’s summation, characters, Dirichlet’s hyperbola -- method),

– The Prime Number Theorem (Chebyshev’s results, \(\zeta(1+it) \not=0\), the Landau-Ikehara theorem -- [called the Ikehara-Wiener theorem in the book], and the prime number theorem),

– The Method of Contour Integration (which gives another proof of the prime number theorem, -- further examples for applications of the method),

– Functional Equations (Poisson’s formula, functional equation for \(\zeta(s)\) and \(L(s,\chi)\)),

– Hadamard Products (Jensen’s theorem, entire functions, zero-free regions),

– Explicit Formulas (for \(\psi(x)\), Weil’s explicit formula),

– The Selberg Class (Dirichlet series \(\sum a_n \cdot n^{-s}\) with a functional equation, with an Euler -- product, and with an estimate \(a_n ={\mathcal O}_{\varepsilon} (n^{\varepsilon})\)),

– Sieve Methods (Eratosthenes’ sieve, Brun’s sieve, Selberg’s sieve; the ‘Large Sieve’ is missing),

– \(p\)-adic Methods (Theorem of Ostrowski, Hensel’s lemma, \(p\)-adic zeta function).

Of course, it is not possible to describe adequately the wealth of material covered in this book. The reviewer thinks that every adept of number theory ought to work by himself through the problems of this monograph – certainly he will gain large benefits.

Reviewer: Wolfgang Schwarz (Frankfurt am Main)

##### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

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

11Nxx | Multiplicative number theory |

11M38 | Zeta and \(L\)-functions in characteristic \(p\) |

11A25 | Arithmetic functions; related numbers; inversion formulas |

11N35 | Sieves |

11A41 | Primes |

30Dxx | Entire and meromorphic functions of one complex variable, and related topics |