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Publisher:

American Mathematical Society

Publication Date:

2006

Number of Pages:

302

Format:

Hardcover

Price:

59.00

ISBN:

0821835807

Category:

Monograph

[Reviewed by , on ]

Fernando Q. Gouvêa

07/17/2006

There's clearly something about Euler that fascinates people. When they are exposed to his work, mathematicians tend to become a little bit obsessed. There is so much of it, it's such a pleasure to read, and it's so rich in ideas and mysterious moves that one sits there amazed at Euler's instinct, abilities, and insight. That's how we end up with books such as Dunham's The Master of Us All (one of the MAA's all-time bestsellers) and columns such as Sandifer's How Euler Did It (one of my favorite *FOCUS Online* columns). Clearly, that is also where Varadarajan's *Euler Through Time* came from. Each of these authors has discovered Euler and wants to tell us about what they found.

Of course, each author takes his own approach. Dunham modernizes the arguments a bit, but tries hard to set Euler in historical context. Sandifer does his best to stay within Euler's own thought world; to my mind, he is the expositor who best displays Euler's own ideas. Varadarajan has chosen to use Euler's work as a jumping off point to range far and wide into mathematics. The result is not quite a history book. Instead, it is a collection of stories about how mathematical ideas evolved and changed from the eighteenth century to our time.

There are two introductory chapters. The first gives a short account of Euler's life. The second, entitled "The Universal Mathematician," is an attempt to survey at least some aspects of Euler's mathematics not discussed in depth in the remainder of the book. Chapters three through six are the meat of the book. Each of them starts with some of Euler's work on infinite series and products, then moves forward in time.

Readers who already know a little about Euler's life and work would do well to simply skip the first two chapters. They are the weakest in the book, both in content and in style. We get clunky statements such as "…he carried in his mind the entire corpus of mathematics and physics of his epoch," learn that Euler's father was a "parish priest" (I suspect the old man would object to that characterization!), and read about Virgil's "*Aeneis*". (Where are the copyeditors of olden days?)

The discussion of Euler's number theory in chapter two is particularly weak and disorganized. For example, we read that "The Babylonians had a deep understanding of Pythagorean triplets" and that Fermat "had great difficulties in writing his arguments down." (The former is definitely false, the latter is a conjecture at best, and one that I do not find very plausible.) The discussion of Fermat primes depends on Fermat's Little Theorem, which is only discussed in the following section of the text, but the discussion, when it comes, omits exactly the crucial fact needed in the previous section. (Specifically, that the order of *x* modulo *m* is a divisor of φ(*m* ).) It would have been easy to put Little Fermat first and to put in that fact. Later, we miss the copyeditor again when we read that Euler worked on "Pells's equation."

Things pick up when we reach the more mathematical chapters. Varadarajan is not the first mathematician to feel the fascination of the material on infinite series and products to be found in volumes 14–16 of *Series Prima* of Euler's *Opera Omnia.* There is something truly special there, and these papers dominate and motivate the rest of the book.

Varadarajan takes up (what we call) special values of the zeta function in chapter three. Chapter four discusses the Euler-Maclaurin summation formula and what can be done with it. The fifth chapter (definitely the best in the book) deals with divergent series, summation methods, and takes that theme all the way to Feynman's path integral approach to quantum mechanics. The final chapter starts with Euler's product formula for the zeta function and chases its implications down to class field theory and the Langlands program.

These chapters vary in how close they stay to Euler's work. Chapters three and four are more historical, though there is no attempt to preserve Euler's notation or approach. In addition to describing Euler's work, Varadarajan often explains how to justify Euler's proofs in modern terms. Chapters five and six start from Euler but leave him fairly quickly to chase the development of the ideas. It is all done very well, particularly in chapter five.

Historians will, of course, be driven nuts by some of this. On page 114, for example, Varadarajan is explaining the origin of the Euler-Maclaurin summation formula and says "Euler's procedure… may be understood very simply using the symbolic method." That suggests that what follows is *not* Euler's procedure (which is never described), but a later interpretation (due to whom?) of it. Similarly, on page 134, he suddenly says "I shall follow Hardy's discussion…", giving us no hint of what *Euler*'s discussion looked like. (Quite different, in fact.) Thus, for someone wanting to understand Euler's work as he understood it, the book will serve mostly as a modern rewriting of what one finds in Euler: clarifying, perhaps, but no substitute for reading the original.

The question that remains is this: who is Varadarajan writing for? A clue is to be found when he says that "As we all know, Wiles solved FLT in 1995." My typical undergraduate students do *not* all know that. The later chapters confirm that Varadarajan expects his readers to have rather broad mathematical knowledge and a high tolerance for passages that run the risk, depending on the reader's background knowledge, of devolving into jabberwocky. It is mostly mathematicians and (fairly advanced) graduate students who will enjoy and learn from this book.

Overall, Varadarajan has provided us with a useful guide to certain portions of Euler's work and with interesting surveys of the mathematics to which that work led over the centuries. I'll certainly refer to it as I try to work through Euler's papers. But I will definitely still need the *Opera Omnia* if I want to see what the man actually did.

Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME. He too has been bitten by the Euler bug.

Preface vii

Chapter 1. Leonhard Euler (1707-1783) 1

1.1. Introduction 1

1.2. Early life 5

1.3. The first stay in St. Petersburg: 1727-1741 8

1.4. The Berlin years: 1741-1766 11

1.5. The second St. Petersburg stay and the last years: 1766-1783 12

1.6. Opera Omnia 13

1.7. The personality of Euler 14

Notes and references 15

Chapter 2. The Universal Mathematician 21

2.1. Introduction 21

2.2. Calculus 21

2.3. Elliptic integrals 23

2.4. Calculus of variations 33

2.5. Number theory 37

Notes and references 57

Chapter 3. Zeta Values 59

3.1. Summary 59

3.2. Some remarks on infinite series and products and their values 64

3.3. Evaluation of *ζ*(2) and *ζ*(4) 68

3.4. Infinite products for circular and hyperbolic functions 77

3.5. The infinite partial fractions for (sin *x*)*−*1 and cot *x*.

Evaluation of *ζ*(2*k*) and *L*(2*k* + 1) 87

3.6. Partial fraction expansions as integrals 94

3.7. Multizeta values 105

Notes and references 110

Chapter 4. Euler-Maclaurin Sum Formula 113

4.1. Formal derivation 113

4.2. The case when the function is a polynomial 116

4.3. Summation formula with remainder terms 117

4.4. Applications 121

Notes and references 124

Chapter 5. Divergent Series and Integrals 125

5.1. Divergent series and Euler’s ideas about summing them 125

5.2. Euler’s derivation of the functional equation of the zeta function 131

v

vi CONTENTS

5.3. Euler’s summation of the factorial series 138

5.4. The general theory of summation of divergent series 145

5.5. Borel summation 152

5.6. Tauberian theorems 158

5.7. Some applications 163

5.8. Fourier integral, Wiener Tauberian theorem, and Gel’fand

transform on commutative Banach algebras 171

5.9. Generalized functions and smeared summation 185

5.10. Gaussian integrals, Wiener measure and the path integral

formulae of Feynman and Kac 191

Notes and references 206

Chapter 6. Euler Products 211

6.1. Euler’s product formula for the zeta function and others 211

6.2. Euler products from Dirichlet to Hecke 217

6.3. Euler products from Ramanujan and Hecke to Langlands 238

6.4. Abelian extensions and class field theory 251

6.5. Artin nonabelian *L*-functions 262

6.6. The Langlands program 264

Notes and references 265

Gallery 269

Sample Pages from *Opera Omnia* 295

Index 301

Preface vii

Chapter 1. Leonhard Euler (1707-1783) 1

1.1. Introduction 1

1.2. Early life 5

1.3. The first stay in St. Petersburg: 1727-1741 8

1.4. The Berlin years: 1741-1766 11

1.5. The second St. Petersburg stay and the last years: 1766-1783 12

1.6. Opera Omnia 13

1.7. The personality of Euler 14

Notes and references 15

Chapter 2. The Universal Mathematician 21

2.1. Introduction 21

2.2. Calculus 21

2.3. Elliptic integrals 23

2.4. Calculus of variations 33

2.5. Number theory 37

Notes and references 57

Chapter 3. Zeta Values 59

3.1. Summary 59

3.2. Some remarks on infinite series and products and their values 64

3.3. Evaluation of *ζ*(2) and *ζ*(4) 68

3.4. Infinite products for circular and hyperbolic functions 77

3.5. The infinite partial fractions for (sin *x*)*−*1 and cot *x*.

Evaluation of *ζ*(2*k*) and *L*(2*k* + 1) 87

3.6. Partial fraction expansions as integrals 94

3.7. Multizeta values 105

Notes and references 110

Chapter 4. Euler-Maclaurin Sum Formula 113

4.1. Formal derivation 113

4.2. The case when the function is a polynomial 116

4.3. Summation formula with remainder terms 117

4.4. Applications 121

Notes and references 124

Chapter 5. Divergent Series and Integrals 125

5.1. Divergent series and Euler’s ideas about summing them 125

5.2. Euler’s derivation of the functional equation of the zeta function 131

v

vi CONTENTS

5.3. Euler’s summation of the factorial series 138

5.4. The general theory of summation of divergent series 145

5.5. Borel summation 152

5.6. Tauberian theorems 158

5.7. Some applications 163

5.8. Fourier integral, Wiener Tauberian theorem, and Gel’fand

transform on commutative Banach algebras 171

5.9. Generalized functions and smeared summation 185

5.10. Gaussian integrals, Wiener measure and the path integral

formulae of Feynman and Kac 191

Notes and references 206

Chapter 6. Euler Products 211

6.1. Euler’s product formula for the zeta function and others 211

6.2. Euler products from Dirichlet to Hecke 217

6.3. Euler products from Ramanujan and Hecke to Langlands 238

6.4. Abelian extensions and class field theory 251

6.5. Artin nonabelian *L*-functions 262

6.6. The Langlands program 264

Notes and references 265

Gallery 269

Sample Pages from *Opera Omnia* 295

Index 301

Preface vii

Chapter 1. Leonhard Euler (1707-1783) 1

1.1. Introduction 1

1.2. Early life 5

1.3. The first stay in St. Petersburg: 1727-1741 8

1.4. The Berlin years: 1741-1766 11

1.5. The second St. Petersburg stay and the last years: 1766-1783 12

1.6. Opera Omnia 13

1.7. The personality of Euler 14

Notes and references 15

Chapter 2. The Universal Mathematician 21

2.1. Introduction 21

2.2. Calculus 21

2.3. Elliptic integrals 23

2.4. Calculus of variations 33

2.5. Number theory 37

Notes and references 57

Chapter 3. Zeta Values 59

3.1. Summary 59

3.2. Some remarks on infinite series and products and their values 64

3.3. Evaluation of *ζ*(2) and *ζ*(4) 68

3.4. Infinite products for circular and hyperbolic functions 77

3.5. The infinite partial fractions for (sin *x*)*−*1 and cot *x*.

Evaluation of *ζ*(2*k*) and *L*(2*k* + 1) 87

3.6. Partial fraction expansions as integrals 94

3.7. Multizeta values 105

Notes and references 110

Chapter 4. Euler-Maclaurin Sum Formula 113

4.1. Formal derivation 113

4.2. The case when the function is a polynomial 116

4.3. Summation formula with remainder terms 117

4.4. Applications 121

Notes and references 124

Chapter 5. Divergent Series and Integrals 125

5.1. Divergent series and Euler’s ideas about summing them 125

5.2. Euler’s derivation of the functional equation of the zeta function 131

v

vi CONTENTS

5.3. Euler’s summation of the factorial series 138

5.4. The general theory of summation of divergent series 145

5.5. Borel summation 152

5.6. Tauberian theorems 158

5.7. Some applications 163

5.8. Fourier integral, Wiener Tauberian theorem, and Gel’fand

transform on commutative Banach algebras 171

5.9. Generalized functions and smeared summation 185

5.10. Gaussian integrals, Wiener measure and the path integral

formulae of Feynman and Kac 191

Notes and references 206

Chapter 6. Euler Products 211

6.1. Euler’s product formula for the zeta function and others 211

6.2. Euler products from Dirichlet to Hecke 217

6.3. Euler products from Ramanujan and Hecke to Langlands 238

6.4. Abelian extensions and class field theory 251

6.5. Artin nonabelian *L*-functions 262

6.6. The Langlands program 264

Notes and references 265

Gallery 269

Sample Pages from *Opera Omnia* 295

Index 301

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