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