Riemann s zeta function (eBook)

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1974 | 1. Auflage
314 Seiten
Elsevier Science (Verlag)
978-0-08-087373-2 (ISBN)

Infos

Riemann?s zeta function

Inhalt

Front Cover 1
Riemann's Zeta Function 4
Copyright Page 5
Contents 6
Preface 10
Acknowledgments 14
Chapter 1. Riemann's Paper 16
1.1 The Historical Context of the Paper 16
1.2 The Euler Product Formula 21
1.3 The Factorial Function 22
1.4 The Function . (s) 24
1.5 Values of . (s) 26
1.6 First Proof of the Functional Equation 27
1.7 Second Proof of the Functional Equation 30
1.8 The Function . (s) 31
1.9 The Roots p of . 33
1.10 The Product Representation of . (s) 33
1.11 The Connection between . (s) and Primes 33
1.12 Fourier Inversion 38
1.13 Method for Deriving the Formula for J (x) 40
1.14 The Principal Term of J ( x) 41
1.15 The Term Involving the Roots p 44
1.16 The Remaining Terms 46
1.17 The Formula for p (x) 48
1.18 The Density dj 51
1.19 Questions Unresolved by Riemann 52
Chapter 2. The Product Formula for 54
2.1 Introduction 54
2.2 Jensen's Theorem 55
2.3 A Simple Estimate of I .(s)I 56
2.4 The Resulting Estimate of the Roots p 57
2.5 Convergence of the Product 57
2.6 Rate of Growth of the Quotient 58
2.7 Rate of Growth of Even Entire Functions 60
2.8 The Product Formula for . 61
Chapter 3. Riemann's Main Formula 63
3.1 Introduction 63
3.2 Derivation of von Mangoldt's Formula for .(x) 65
3.3 The Basic Integral Formula 69
3.4 The Density of the Roots 71
3.5 Proof of von Mangoldt's Formula for . (x) 73
3.6 Riemann's Main Formula 76
3.7 Von Mangoldt’s Proof of Riemann’s Main Formula 77
3.8 Numerical Evaluation of the Constant 81
Chapter 4. The Prime Number Theorem 83
4.1 Introduction 83
4.2 Hadamard's Proof That Re p < 1 for All p
4.3 Proof That .(x)~x 87
4.4 Proof of the Prime Number Theorem 91
Chapter 5. De la Vall6e Poussin's Theorem 93
5.1 Introduction 93
5.2 An Improvement of Re p < 1
5.3 De la Vallte Poussin's Estimate of the Error 96
5.4 Other Formulas for R(X) 99
5.5 Error Estimates and the Riemann Hypothesis 103
5.6 A Postscript to de la Vallte Poussin's Proof 106
Chapter 6. Numerical Analysis of the Roots by Euler-Maclaurin Summation 111
6.1 Introduction 111
6.2 Euler-Maclaurin Summation 113
6.3 Evaluation of II by Euler-Maclaurin Summation. Stirling's Series 121
6.4 Evaluation of 3 by Euler-Maclaurin Summation 129
6.5 Techniques for Locating Roots on the Line 134
6.6 Techniques for Computing the Number of Roots in a Given Range 142
6.7 Backlund's Estimate of N (T) 147
6.8 Alternative Evaluation of .'(0)/.(0) 149
Chapter 7. The Riemann-Siege1 Formula 151
7.1 Introduction 151
7.2 Basic Derivation of the Formula 152
7.3 Estimation of the Integral away from the Saddle Point 156
7.4 First Approximation to the Main Integral 160
7.5 Higher Order Approximations 163
7.6 Sample Computations 170
7.7 Error Estimates 177
7.8 Speculations on the Genesis of the Riemann Hypothesis 179
7.9 The Riemann-Siege1 Integral Formula 181
Chapter 8. Large-Scale Computations 186
8.1 Introduction 186
8.2 Turing's Method 187
8.3 Lehmer's Phenomenon 190
8.4 Computations of Rosser, Yohe, and Schoenfeld 194
Chapter 9. The Growth of Zeta as t .8 and the Location of Its Zeros 197
9.1 Introduction 197
9.2 Lindelöf's Estimates and His Hypothesis 198
9.3 The Three Circles Theorem 202
9.4 Backlund's Reformulation of the Lindelöf Hypothesis 203
9.5 The Average Value of S ( t ) Is Zero 205
9.6 The Bohr-Landau Theorem 208
9.7 The Average of . (s)/2 210
9.8 Further Results. Landau's Notation 0, 0 214
Chapter 10. Fourier Analysis 218
10.1 Invariant Operators on R+ and Their Transforms 218
10.2 Adjoints and Their Transforms 220
10.3 A Self-Adjoint Operator with Transform .(s) 221
10.4 The Functional Equation 224
10.5 2.( s )/s(s–1) as a Transform 227
10.6 Fourier Inversion 228
10.7 Parseval's Equation 230
10.8 The Values of (—n) 231
10.9 Möbius Inversion 232
10.10 Ramanujan's Formula 233
Chapter 11. Zeros on the Line 241
11.1 Hardy's Theorem 241
11.2 There Are at Least KT Zeros on the Line 244
11.3 There Are at Least KT log T Zeros on the Line 252
11.4 Proof of a Lemma 261
Chapter 12. Miscellany 275
12.1 The Riemann Hypothesis and the Growth of M (x) 275
12.2 The Riemann Hypothesis and Farey Series 278
12.3 Denjoy's Probabilistic Interpretation of the Riemann Hypothesis 283
12.4 An Interesting False Conjecture 284
12.5 Transforms with Zeros on the Line 284
12.6 Alternative Proof of the Integral Formula 288
12.7 Tauberian Theorems 293
12.8 Chebyshev's Identity 296
12.9 Selberg's Inequality 299
12.10 Elementary Proof of the Prime Number Theorem 303
12.11 Other Zeta Functions. Weil's Theorem 313
Appendix On the Number of Primes Less Than a Given Magnitude (By Bernhard Riemann) 314
References 321
Index 326

Produkt-Details

• Erscheint lt. Verlag: 31.5.1974
• Mitarbeit: Herausgeber (Serie): H.M. Edwards
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Technik
• ISBN-10: 0-08-087373-1 / 0080873731
• ISBN-13: 978-0-08-087373-2 / 9780080873732