Neverending fractions. An introduction to continued fractions.

*(English)*Zbl 1307.11001
Australian Mathematical Society Lecture Series 23. Cambridge: Cambridge University Press (ISBN 978-0-521-18649-0/pbk; 978-1-107-00665-2/hbk; 978-0-511-90265-9/ebook). x, 212 p. (2014).

Continued fractions form a classical area within number theory, the roots of which can be traced back to Euclid’s algorithm for the greatest common divisor of two integers (300 BC). Several centuries ago, Rafael Bombelli (1579), Pietro Cataldi (1613), and John Wallis (1695) developed the method of continued fractions for rational approximations of irrational numbers (such as square roots), and later on great mathematicians like Leonhard Euler (1737 and 1748), Johann Lambert (1761), Joseph L. Lagrange (1768 and 1770), Carl Friedrich Gauss (1813), and others discovered various fundamental properties and important applications of continued fractions. In fact, these fascinating objects have been a very active field of research ever since, and the vast contemporary literature on continued fractions evidently shows that this topic is still far from being exhausted.

The book under review grew out of many lectures that the four authors delivered independently on different occasions to students of different levels. Its main goal is to provide an introduction to continued fractions for a wide audience of readers, including graduate students, postgraduates, researchers as well as teachers and even amateurs in mathematics. As the authors point out in the preface, their intention is to demonstrate that continued fractions represent a neverending research field, with a wealth of results elementary enough to be explained to this target readership.

Regarding the precise contents, the book comprises nine chapters, each of which is divided into several sections. While the first three chapters are devoted to a general introduction to continued fractions, the subsequent six chapters deal with more special topics and applications of the theory.

Chapter 1 presents the necessary prerequisites from elementary number theory with full proofs. These concern the following themes: divisibility of integers and the Euclidean algorithm, prime numbers and the fundamental theorem of arithmetic, Fibonacci numbers and the complexity of the Euclidean algorithm, approximation of real numbers by rationals and Farey sequences. Chapter 2 begins the study of continued fractions and their algebraic theory, thereby explaining the continued fraction of a real number in general, the principle of Diophantine approximation, the continued fraction of a quadratic irrational and the Euler-Lagrange theorem in this context, the construction of real numbers with bounded partial quotients, and other results on rational approximation.

Chapter 3 touches upon the metric theory of continued fractions, with emphasis on the growth of partial quotients of a continued fraction of a real number, the approximation of almost all real numbers by rationals, and the classical Gauss-Kuzmin statistics in metric number theory. Chapters 4, 5 and 6 originate in lectures that one of the authors, the late Alf van der Poorten (1942–2010), gave in the last few years before his untimely death. Chapter 4 is titled “Quadratic irrationals through a magnifier” and contains some informal lectures on continued fractions of algebraic numbers, Pell’s equation, and some concrete examples.

Chapter 5 is a survey of aspects of continued fractions in function fields, with a view toward some so-called (recursively defined) Somos sequences, pseudo-elliptic integrals, and hyperelliptic curves, whereas Chapter 6 briefly discusses the relationship between neverending paper foldings and continued fractions. Chapter 7 provides the study of a class of generating functions that are connected to remarkable continued fractions and rational approximations. Lambert series expansions of generating functions and an inhomogeneous Diophantine approximation algorithm are the main tools applied here.

Chapter 8 treats the Erdős-Moser equation \[ 1^k+ 2^k+\cdots+ (m-2)^k+ (m-1)^k= m^k \] and its possible integer solutions for \(m\geq 2\) and \(k\geq 2\)

A conjecture by P. Erdős states that such solutions do not exist, and L. Moser proved in 1953 that only for even exponents \(k\) and rather large integers \(m\) such solutions could be expected at all.

In this chapter, both the arithmetic and the analysis of the Erdős-Moser equation are outlined, where efficient ways of computing certain associated continued fractions as well as explicit bounds for solutions are presented. The basic reference for this chapter is the recent paper by Y. Gallot et al. [Math. Comput. 80, No. 274, 1221–1237 (2011; Zbl 1231.11038)].

The concluding Chapter 9 finally turns to irregular continued fractions by surveying their general theory as well as some important examples, including Gauss’ irregular continued fraction for the hypergeometric function, Ramanujan’s arithmetic-geometric mean (AGM) continued fraction (from his second notebook) and related developments by one of the authors of the present book ( Borwein) and his collaborators, an irregular continued fraction for the zeta value \(\zeta(2)= \pi^2/6\), and a new proof of R. Apéry’s theorem on the irrationality of \(\zeta(3)\) [Astérisque 61, 11–13 (1979; Zbl 0401.10049)] as a striking application of the foregoing discussion.

There is an appendix to the main text containing a collection of interesting continued fractions, both regular and irregular, where most of those represent special real numbers, values of special functions, particular infinite series, and some \(q\)-series, respectively.

As one can see, the book is a combination of formal and informal styles of expository writing, and a mixture of introductory textbook and topical surveys likewise. Many of the special topics discussed in the later chapters are not to be found in other books but only in scattered articles and lectures. As for full details with regard to these topics chapters, the reader is referred to the original research papers listed in the rich bibliography. In fact, each chapter ends with a set of notes providing additional remarks and hints for further reading, and a few exercises invite the reader to acquire complementary knowledge through independent work.

All together, the present book gives a beautiful panoramic view of the “neverending story of neverending fractions” by making apparent their naturalness, their ubiquity, and their wide-range of applications in very lucid and inspiring a manner.

The book under review grew out of many lectures that the four authors delivered independently on different occasions to students of different levels. Its main goal is to provide an introduction to continued fractions for a wide audience of readers, including graduate students, postgraduates, researchers as well as teachers and even amateurs in mathematics. As the authors point out in the preface, their intention is to demonstrate that continued fractions represent a neverending research field, with a wealth of results elementary enough to be explained to this target readership.

Regarding the precise contents, the book comprises nine chapters, each of which is divided into several sections. While the first three chapters are devoted to a general introduction to continued fractions, the subsequent six chapters deal with more special topics and applications of the theory.

Chapter 1 presents the necessary prerequisites from elementary number theory with full proofs. These concern the following themes: divisibility of integers and the Euclidean algorithm, prime numbers and the fundamental theorem of arithmetic, Fibonacci numbers and the complexity of the Euclidean algorithm, approximation of real numbers by rationals and Farey sequences. Chapter 2 begins the study of continued fractions and their algebraic theory, thereby explaining the continued fraction of a real number in general, the principle of Diophantine approximation, the continued fraction of a quadratic irrational and the Euler-Lagrange theorem in this context, the construction of real numbers with bounded partial quotients, and other results on rational approximation.

Chapter 3 touches upon the metric theory of continued fractions, with emphasis on the growth of partial quotients of a continued fraction of a real number, the approximation of almost all real numbers by rationals, and the classical Gauss-Kuzmin statistics in metric number theory. Chapters 4, 5 and 6 originate in lectures that one of the authors, the late Alf van der Poorten (1942–2010), gave in the last few years before his untimely death. Chapter 4 is titled “Quadratic irrationals through a magnifier” and contains some informal lectures on continued fractions of algebraic numbers, Pell’s equation, and some concrete examples.

Chapter 5 is a survey of aspects of continued fractions in function fields, with a view toward some so-called (recursively defined) Somos sequences, pseudo-elliptic integrals, and hyperelliptic curves, whereas Chapter 6 briefly discusses the relationship between neverending paper foldings and continued fractions. Chapter 7 provides the study of a class of generating functions that are connected to remarkable continued fractions and rational approximations. Lambert series expansions of generating functions and an inhomogeneous Diophantine approximation algorithm are the main tools applied here.

Chapter 8 treats the Erdős-Moser equation \[ 1^k+ 2^k+\cdots+ (m-2)^k+ (m-1)^k= m^k \] and its possible integer solutions for \(m\geq 2\) and \(k\geq 2\)

A conjecture by P. Erdős states that such solutions do not exist, and L. Moser proved in 1953 that only for even exponents \(k\) and rather large integers \(m\) such solutions could be expected at all.

In this chapter, both the arithmetic and the analysis of the Erdős-Moser equation are outlined, where efficient ways of computing certain associated continued fractions as well as explicit bounds for solutions are presented. The basic reference for this chapter is the recent paper by Y. Gallot et al. [Math. Comput. 80, No. 274, 1221–1237 (2011; Zbl 1231.11038)].

The concluding Chapter 9 finally turns to irregular continued fractions by surveying their general theory as well as some important examples, including Gauss’ irregular continued fraction for the hypergeometric function, Ramanujan’s arithmetic-geometric mean (AGM) continued fraction (from his second notebook) and related developments by one of the authors of the present book ( Borwein) and his collaborators, an irregular continued fraction for the zeta value \(\zeta(2)= \pi^2/6\), and a new proof of R. Apéry’s theorem on the irrationality of \(\zeta(3)\) [Astérisque 61, 11–13 (1979; Zbl 0401.10049)] as a striking application of the foregoing discussion.

There is an appendix to the main text containing a collection of interesting continued fractions, both regular and irregular, where most of those represent special real numbers, values of special functions, particular infinite series, and some \(q\)-series, respectively.

As one can see, the book is a combination of formal and informal styles of expository writing, and a mixture of introductory textbook and topical surveys likewise. Many of the special topics discussed in the later chapters are not to be found in other books but only in scattered articles and lectures. As for full details with regard to these topics chapters, the reader is referred to the original research papers listed in the rich bibliography. In fact, each chapter ends with a set of notes providing additional remarks and hints for further reading, and a few exercises invite the reader to acquire complementary knowledge through independent work.

All together, the present book gives a beautiful panoramic view of the “neverending story of neverending fractions” by making apparent their naturalness, their ubiquity, and their wide-range of applications in very lucid and inspiring a manner.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

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

11A55 | Continued fractions |

11J70 | Continued fractions and generalizations |

11K50 | Metric theory of continued fractions |

11Y65 | Continued fraction calculations (number-theoretic aspects) |