Number theory. An introduction.

*(English)*Zbl 0847.11001
Pure and Applied Mathematics, Marcel Dekker. 201. Basel: Marcel Dekker. xii, 749 p. (1996).

At the beginning of the twentieth century there were only a few textbooks on number theory. Today there are several hundred. The bibliography at the end of the text under review lists seventy titles published in the twentieth century, and the list is far from complete. Because of the explosive growth of the subject there has been a recent tendency to enlarge considerably the scope of introductory books on number theory.

The author of this book has given a wide interpretation of his title. The table of contents suggests that a more appropriate title might have been “An introduction to some topics in elementary number theory”. The topics are divided among ten chapters as follows: (1) Primes and divisibility, including unique factorization and the Euclidean algorithm; (2) Congruences, including primitive roots, applications to cryptography, and pseudoprimes; (3) Quadratic residues and the quadratic reciprocity law; (4) Diophantine approximation, including Farey sequences and continued fractions; (5) Diophantine equations, with special emphasis on Pythagorean triples and related quadratic equations; (6) More Diophantine problems, including linear equations and sums of squares; (7) Arithmetical functions, with emphasis on multiplicative functions such as the divisor functions, Euler’s totient, and Dirichlet characters. Although the Dirichlet convolution is used to derive identities, there is no mention in this chapter of Dirichlet series as generating functions; (8) The average order of magnitude of arithmetical functions. This chapter devotes a dozen pages developing the Riemann-Stieltjes integral, which is then used to derive the Euler-Maclaurin summation formula; (9) Prime number theory – specifically, an elementary proof of the prime number theorem based on the treatment in the third edition of G. H. Hardy and E. M. Wright [An introduction to the theory of numbers, 3rd ed. (Clarendon Press 1954; Zbl 0058.03301)], and an elementary proof of Dirichlet’s theorem on primes in arithmetical progressions; (10) A brief introduction to algebraic number theory, with special emphasis on quadratic number fields.

This work is obviously a labor of love, despite the hidden message in Problem 7 on page 721. The author has chosen a wide variety of topics that illustrate many different techniques used in number theory, and he succeeds in showing how other parts of mathematics play a role in studying properties of the integers. One may not always agree with the choice of topics or the manner of presentation, but these are matters of personal taste.

The back cover describes the book as a reference for research mathematicians interested in algebra and number theory, and a text for upper-level undergraduates and graduate students in these disciplines. Undergraduates not acquainted with criteria for convergence of infinite products may be baffled by a statement on page 613 concerning an infinite product being positive.

The book contains several hundred exercises, distributed among some sixty problem sets, with additional problems at the end of each chapter, plus further problems requiring the use of a computer. It also contains four tables listing primes, primitive roots, values of some arithmetical functions, and continued fractions of irrational square roots, with solutions of associated Pell equations. And there is a lengthy bibliography and an index that includes a list of mathematical notations under the entry Symbol. But some symbols are not defined anywhere in the book, for example, \(\mathbb{N}\) on page 5, \(\mathbb{Z}\) on page 7, \(\mathbb{Z}_+\) on page 10, and \(\Gamma\) on page 560. It is not clear what criteria were used for including names in the index. For example, an obscure mathematician G. Métrod is listed as being responsible for a little known identity on page 501, but neither Brauer nor Rademacher is listed for the better known Brauer-Rademacher identity on page 498.

The Preface states that many results are given as problem sets, with reference to sources. However, original sources are mentioned only rarely. For example, Harold N. Shapiro is not credited for his Tauberian Theorem included in Problem 5 on page 626, nor is Basil Gordon for his profound results in Problems 8 through 10 on pages 627 and 628, which, in effect, outline an elementary proof of the prime number theorem.

This reviewer was puzzled by the inclusion of certain topics in the problem sets. For example, Dedekind sums are defined in Problem 3 on page 562, without motivation or explanation of why they are worth studying, and the reader is asked to derive all their principal properties as exercises, including the non-trivial reciprocity law.

Every book contains errors, and this one is no exception. The most serious lapse found by the reviewer occurs in the proof of the prime number theorem given in Chapter 9. The relation \(\int^\zeta_0 o(\xi) d\xi= o(\zeta^2)\) used at the top of page 591, although correct, needs some justification because the dummy variable \(\xi\) takes small values as well as large values. On page 9 the symbol \(a\mid b\) is defined only if \(a\) is nonzero, but Theorem 1.2 (b) states that \(0\mid 0\). On page 62 the definition of \(\varphi(m)\) is incomplete because it does not cover the case \(m= 1\). On page 71, Wilson’s theorem as stated is false when \(n= 1\).

There are many misprints that should have been caught by an alert copy editor. On page 3, the spelling of de la Vallée Poussin’s name contains two errors that are repeated on page 739. The names of Kloosterman, von Mangoldt and Wolstenholme are also spelled incorrectly. On pages 24-28, there are ten citations to the Bibliography, each with an incorrect reference number.

In conclusion, this reviewer feels that a 750-page book containing hundreds of problems, many of which a novice will not be able to solve, may intimidate many beginners who are fascinated by number theory and wish to acquire an introduction to its concepts and techniques. This book could be made more appealing by judicious pruning.

The author of this book has given a wide interpretation of his title. The table of contents suggests that a more appropriate title might have been “An introduction to some topics in elementary number theory”. The topics are divided among ten chapters as follows: (1) Primes and divisibility, including unique factorization and the Euclidean algorithm; (2) Congruences, including primitive roots, applications to cryptography, and pseudoprimes; (3) Quadratic residues and the quadratic reciprocity law; (4) Diophantine approximation, including Farey sequences and continued fractions; (5) Diophantine equations, with special emphasis on Pythagorean triples and related quadratic equations; (6) More Diophantine problems, including linear equations and sums of squares; (7) Arithmetical functions, with emphasis on multiplicative functions such as the divisor functions, Euler’s totient, and Dirichlet characters. Although the Dirichlet convolution is used to derive identities, there is no mention in this chapter of Dirichlet series as generating functions; (8) The average order of magnitude of arithmetical functions. This chapter devotes a dozen pages developing the Riemann-Stieltjes integral, which is then used to derive the Euler-Maclaurin summation formula; (9) Prime number theory – specifically, an elementary proof of the prime number theorem based on the treatment in the third edition of G. H. Hardy and E. M. Wright [An introduction to the theory of numbers, 3rd ed. (Clarendon Press 1954; Zbl 0058.03301)], and an elementary proof of Dirichlet’s theorem on primes in arithmetical progressions; (10) A brief introduction to algebraic number theory, with special emphasis on quadratic number fields.

This work is obviously a labor of love, despite the hidden message in Problem 7 on page 721. The author has chosen a wide variety of topics that illustrate many different techniques used in number theory, and he succeeds in showing how other parts of mathematics play a role in studying properties of the integers. One may not always agree with the choice of topics or the manner of presentation, but these are matters of personal taste.

The back cover describes the book as a reference for research mathematicians interested in algebra and number theory, and a text for upper-level undergraduates and graduate students in these disciplines. Undergraduates not acquainted with criteria for convergence of infinite products may be baffled by a statement on page 613 concerning an infinite product being positive.

The book contains several hundred exercises, distributed among some sixty problem sets, with additional problems at the end of each chapter, plus further problems requiring the use of a computer. It also contains four tables listing primes, primitive roots, values of some arithmetical functions, and continued fractions of irrational square roots, with solutions of associated Pell equations. And there is a lengthy bibliography and an index that includes a list of mathematical notations under the entry Symbol. But some symbols are not defined anywhere in the book, for example, \(\mathbb{N}\) on page 5, \(\mathbb{Z}\) on page 7, \(\mathbb{Z}_+\) on page 10, and \(\Gamma\) on page 560. It is not clear what criteria were used for including names in the index. For example, an obscure mathematician G. Métrod is listed as being responsible for a little known identity on page 501, but neither Brauer nor Rademacher is listed for the better known Brauer-Rademacher identity on page 498.

The Preface states that many results are given as problem sets, with reference to sources. However, original sources are mentioned only rarely. For example, Harold N. Shapiro is not credited for his Tauberian Theorem included in Problem 5 on page 626, nor is Basil Gordon for his profound results in Problems 8 through 10 on pages 627 and 628, which, in effect, outline an elementary proof of the prime number theorem.

This reviewer was puzzled by the inclusion of certain topics in the problem sets. For example, Dedekind sums are defined in Problem 3 on page 562, without motivation or explanation of why they are worth studying, and the reader is asked to derive all their principal properties as exercises, including the non-trivial reciprocity law.

Every book contains errors, and this one is no exception. The most serious lapse found by the reviewer occurs in the proof of the prime number theorem given in Chapter 9. The relation \(\int^\zeta_0 o(\xi) d\xi= o(\zeta^2)\) used at the top of page 591, although correct, needs some justification because the dummy variable \(\xi\) takes small values as well as large values. On page 9 the symbol \(a\mid b\) is defined only if \(a\) is nonzero, but Theorem 1.2 (b) states that \(0\mid 0\). On page 62 the definition of \(\varphi(m)\) is incomplete because it does not cover the case \(m= 1\). On page 71, Wilson’s theorem as stated is false when \(n= 1\).

There are many misprints that should have been caught by an alert copy editor. On page 3, the spelling of de la Vallée Poussin’s name contains two errors that are repeated on page 739. The names of Kloosterman, von Mangoldt and Wolstenholme are also spelled incorrectly. On pages 24-28, there are ten citations to the Bibliography, each with an incorrect reference number.

In conclusion, this reviewer feels that a 750-page book containing hundreds of problems, many of which a novice will not be able to solve, may intimidate many beginners who are fascinated by number theory and wish to acquire an introduction to its concepts and techniques. This book could be made more appealing by judicious pruning.

Reviewer: T.M.Apostol (Pasadena)

##### MSC:

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

11Dxx | Diophantine equations |

11Axx | Elementary number theory |