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(2k) and L(2k + 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(2k) and L(2k + 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(2k) and L(2k + 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