##
**Euler through time. A new look at old themes.**
*(English)*
Zbl 1096.01013

Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3580-7/hbk). viii, 302 p. (2006).

This book grew out of a course on the history of mathematics that the author, a highly renowned researcher, teacher, and textbook author in various fields of contemporary mathematics, has taught at the University of California at Los Angeles (UCLA) during the winter quarter of 2001. Instead of following the standard way of teaching such a course by starting with Babylonian mathematics, proceeding then with the following ancient developments in number theory, geometry and algebra, analyzing the genesis of infinitesimal calculus, and ending up with the great mathematical achievements of the eighteenth and nineteenth century, the author decided to do it differently and focus on the work of a single great figure in the history of mathematics. In his opinion, the history of mathematics is too important (and complex) to be left entirely to historians, and a complete mathematical history should also analyze the evolution of pioneering ideas and how they mesh with current mathematics. In this vein, the author chose Leonhard Euler and parts of his epoch-making work in the eighteenth century as the subject of a profound historical analysis of (parts of) contemporary mathematics.

In fact, Leonhard Euler was certainly the greatest mathematician and natural philosopher of the eighteenth century and one of the greatest of all time. As Laplace once put it, all the mathematicians of his time were Euler’s students. Working on all branches of mathematics and physics known in his time, Euler’s universality remained unparalleled ever since, and his enormous contributions to the genesis and development of modern mathematics were absolutely unique until today. He created new branches of mathematics, like combinatorial topology, graph theory, and the calculus of variations; he laid the foundations of modern infinitesimal calculus as it is taught and used today; he started the process of establishing number theory as a major discipline in mathematics; he wrote the first generally accessible books on all these topics, and he launched mathematical research programs for the following two centuries. Anticipating his numerous great successors by more than a century, Euler began theories whose full development by his successors took hundreds of years, and his ingenious discoveries would require for their eventual understanding many sophisticated developments in twentieth-century mathematics.

The book under review is not just one of the many historical essays on the most profilic mathematician of all time. Its main goal is to discuss some of Euler’s versatile themes, methods and results, and how they effect the current research in mathematics from a modern perspective. As the range and volume of Euler’s output is so much imposing, including over 850 substantial papers and more than 25 books, the author had to make a reasonable selection. Thus he has limited himself mostly to one of Euler’s greatest masterpieces, namely his pioneering work on infinite (divergent) series and infinite products, together with its consequences for current research topics like the theory of zeta values, divergent series and integrals, Feynman path integrals in physics, class field theory, and the Langlands program in modern number theory. The book contains six chapters, each of which is divided into up to ten sections.

Chapter 1 is devoted to the biography of Leonhard Euler (1707–1783), including his early life in Basel, Switzerland, his studies and work in mathematics with Johann Bernoulli, his stages of life in St. Petersburg, Russia, and Berlin, Prussia, his relations with his colleagues and royal patrons there, his personality both as a human being and a mathematical researcher, and a short comment on his famous “Opera omnia”, the collection of all of his writings edited by the Euler Commission (founded in 1907) of the Swiss Academy of Sciences.

Chapter 2, entitled “The Universal Mathematician”, is still merely historical and sketches briefly Euler’s work on those themes that were to grow into new areas of mathematics, in the hands of his successors, and which are still flourishing in our times: differential and integral calculus, elliptic integrals, the calculus of variations, and the theory of numbers beyond Fermat.

Chapter 3 carefully explains Euler’s creation of a first systematic and general theory of infinite series, and how he used it to compute, in anticipation of what turned out to be crucial hundreds of years thereafter, several values of the Riemann zeta function. Also, Euler’s idea of representing circular and hyperbolic functions by infinite products is carried out in detail, and his invention of multi-zeta series is discussed at the end of this chapter. In view of their close relationships to modern number theory, topology, and algebraic geometry, multi-zeta values and their (conjectural) identities are of tremendous interest in current research.

Chapter 4 treats Euler’s approach to the problem of summability of function values, culminating in the derivation of the celebrated Euler-Maclaurin sum formula and related concepts such as Bernoulli numbers, summation formulae with remainder terms, the Euler-Mascheroni constant, and Stirling’s formula.

Chapter 5 turns to Euler’s work on divergent series and integrals. His general theory of summation of divergent series is decribed thoroughly, and the vast developments initiated by Euler’s pioneering work are discussed thereafter. This includes Borel summation, Tauberian theorems, Fourier integrals, Wiener’s Tauberian theorem, the Gel’fand transform on commutative Banach algebras, a proof of the prime number theorem, generalized functions and smeared summation, Gaussian integrals, the Wiener measure, the path integral formulae of Feynman and Kac, and other topical themes in current mathematics. In the course of his exposition, the author has set high value on demonstrating how Euler’s work is still penetrating many actual developments today, and to what enormous extent his successors have always profited from his ingenious ideas.

Finally, Chapter 6 gives a brief account of the theory of Euler products and its ubiquity in modern algebraic number theory. Starting from Euler’s product formulae for the zeta function and other number-theoretic functions, the author explains the significance of Euler products in the works of Dirichlet, Riemann, Hurwitz, Dedekind, and Hensel, with an outlook to the more recent developments in algebraic number theory that led to class field theory à la Chevalley, Artin, Tate, Takagi, and others. Euler products and their significance in the works of Ramanujam, Hecke, Petersson, Jacquet, Langlands, Harish-Chandra, and other twentieth-century mathematicians are discussed in the sequel, followed by a discourse on abelian field extensions, Kronecker’s Jugendtraum, Artin’s reciprocity law in class field theory, Artin’s non-abelian \(L\)-functions, and the fascinating Langlands program in current algebraic number theory.

Altogether, the author’s historical treatise leads the reader from Leonhard Euler’s pioneering ideas and contributions in the eighteenth century through a period of 250 years, up to the forefront of current research in algebraic number theory. The progression of ideas regarding divergent series and integrals from Euler’s work to various parts of modern analysis, arithmetic, algebraic geometry, and theoretical physics is exhibited in a unique and masterly manner, thereby highly enlightening and clarifying even for specialists. Despite the advanced material in the later chapters, the author has admirably managed to organize the text in such a manner that an interested non-specialist will find the whole story comprehensible, absorbing and enjoyable.

This book has been written with the greatest insight, expertise, experience, and passion on the part of the author’s, and it should be seen as what it really is: a cultural jewel in the mathematical literature as a whole!

In fact, Leonhard Euler was certainly the greatest mathematician and natural philosopher of the eighteenth century and one of the greatest of all time. As Laplace once put it, all the mathematicians of his time were Euler’s students. Working on all branches of mathematics and physics known in his time, Euler’s universality remained unparalleled ever since, and his enormous contributions to the genesis and development of modern mathematics were absolutely unique until today. He created new branches of mathematics, like combinatorial topology, graph theory, and the calculus of variations; he laid the foundations of modern infinitesimal calculus as it is taught and used today; he started the process of establishing number theory as a major discipline in mathematics; he wrote the first generally accessible books on all these topics, and he launched mathematical research programs for the following two centuries. Anticipating his numerous great successors by more than a century, Euler began theories whose full development by his successors took hundreds of years, and his ingenious discoveries would require for their eventual understanding many sophisticated developments in twentieth-century mathematics.

The book under review is not just one of the many historical essays on the most profilic mathematician of all time. Its main goal is to discuss some of Euler’s versatile themes, methods and results, and how they effect the current research in mathematics from a modern perspective. As the range and volume of Euler’s output is so much imposing, including over 850 substantial papers and more than 25 books, the author had to make a reasonable selection. Thus he has limited himself mostly to one of Euler’s greatest masterpieces, namely his pioneering work on infinite (divergent) series and infinite products, together with its consequences for current research topics like the theory of zeta values, divergent series and integrals, Feynman path integrals in physics, class field theory, and the Langlands program in modern number theory. The book contains six chapters, each of which is divided into up to ten sections.

Chapter 1 is devoted to the biography of Leonhard Euler (1707–1783), including his early life in Basel, Switzerland, his studies and work in mathematics with Johann Bernoulli, his stages of life in St. Petersburg, Russia, and Berlin, Prussia, his relations with his colleagues and royal patrons there, his personality both as a human being and a mathematical researcher, and a short comment on his famous “Opera omnia”, the collection of all of his writings edited by the Euler Commission (founded in 1907) of the Swiss Academy of Sciences.

Chapter 2, entitled “The Universal Mathematician”, is still merely historical and sketches briefly Euler’s work on those themes that were to grow into new areas of mathematics, in the hands of his successors, and which are still flourishing in our times: differential and integral calculus, elliptic integrals, the calculus of variations, and the theory of numbers beyond Fermat.

Chapter 3 carefully explains Euler’s creation of a first systematic and general theory of infinite series, and how he used it to compute, in anticipation of what turned out to be crucial hundreds of years thereafter, several values of the Riemann zeta function. Also, Euler’s idea of representing circular and hyperbolic functions by infinite products is carried out in detail, and his invention of multi-zeta series is discussed at the end of this chapter. In view of their close relationships to modern number theory, topology, and algebraic geometry, multi-zeta values and their (conjectural) identities are of tremendous interest in current research.

Chapter 4 treats Euler’s approach to the problem of summability of function values, culminating in the derivation of the celebrated Euler-Maclaurin sum formula and related concepts such as Bernoulli numbers, summation formulae with remainder terms, the Euler-Mascheroni constant, and Stirling’s formula.

Chapter 5 turns to Euler’s work on divergent series and integrals. His general theory of summation of divergent series is decribed thoroughly, and the vast developments initiated by Euler’s pioneering work are discussed thereafter. This includes Borel summation, Tauberian theorems, Fourier integrals, Wiener’s Tauberian theorem, the Gel’fand transform on commutative Banach algebras, a proof of the prime number theorem, generalized functions and smeared summation, Gaussian integrals, the Wiener measure, the path integral formulae of Feynman and Kac, and other topical themes in current mathematics. In the course of his exposition, the author has set high value on demonstrating how Euler’s work is still penetrating many actual developments today, and to what enormous extent his successors have always profited from his ingenious ideas.

Finally, Chapter 6 gives a brief account of the theory of Euler products and its ubiquity in modern algebraic number theory. Starting from Euler’s product formulae for the zeta function and other number-theoretic functions, the author explains the significance of Euler products in the works of Dirichlet, Riemann, Hurwitz, Dedekind, and Hensel, with an outlook to the more recent developments in algebraic number theory that led to class field theory à la Chevalley, Artin, Tate, Takagi, and others. Euler products and their significance in the works of Ramanujam, Hecke, Petersson, Jacquet, Langlands, Harish-Chandra, and other twentieth-century mathematicians are discussed in the sequel, followed by a discourse on abelian field extensions, Kronecker’s Jugendtraum, Artin’s reciprocity law in class field theory, Artin’s non-abelian \(L\)-functions, and the fascinating Langlands program in current algebraic number theory.

Altogether, the author’s historical treatise leads the reader from Leonhard Euler’s pioneering ideas and contributions in the eighteenth century through a period of 250 years, up to the forefront of current research in algebraic number theory. The progression of ideas regarding divergent series and integrals from Euler’s work to various parts of modern analysis, arithmetic, algebraic geometry, and theoretical physics is exhibited in a unique and masterly manner, thereby highly enlightening and clarifying even for specialists. Despite the advanced material in the later chapters, the author has admirably managed to organize the text in such a manner that an interested non-specialist will find the whole story comprehensible, absorbing and enjoyable.

This book has been written with the greatest insight, expertise, experience, and passion on the part of the author’s, and it should be seen as what it really is: a cultural jewel in the mathematical literature as a whole!

Reviewer: Werner Kleinert (Berlin)

### MSC:

01A70 | Biographies, obituaries, personalia, bibliographies |

01A50 | History of mathematics in the 18th century |

40-03 | History of sequences, series, summability |

11-03 | History of number theory |

11Bxx | Sequences and sets |

11Sxx | Algebraic number theory: local fields |