UID:
almafu_9958094984202883
Umfang:
1 online resource (331 p.)
ISBN:
1-281-76344-6
,
9786611763442
,
0-08-087373-1
Serie:
Pure and applied mathematics; a series of monographs and textbooks ; 58
Inhalt:
RiemannÆs zeta function
Anmerkung:
Description based upon print version of record.
,
Front Cover; Riemann's Zeta Function; Copyright Page; Contents; Preface; Acknowledgments; Chapter 1. Riemann's Paper; 1.1 The Historical Context of the Paper; 1.2 The Euler Product Formula; 1.3 The Factorial Function; 1.4 The Function ζ (s); 1.5 Values of ζ (s); 1.6 First Proof of the Functional Equation; 1.7 Second Proof of the Functional Equation; 1.8 The Function ζ (s); 1.9 The Roots p of ζ; 1.10 The Product Representation of ζ (s); 1.11 The Connection between ζ (s) and Primes; 1.12 Fourier Inversion; 1.13 Method for Deriving the Formula for J (x); 1.14 The Principal Term of J ( x)
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1.15 The Term Involving the Roots p1.16 The Remaining Terms; 1.17 The Formula for π (x); 1.18 The Density dj; 1.19 Questions Unresolved by Riemann; Chapter 2. The Product Formula for; 2.1 Introduction; 2.2 Jensen's Theorem; 2.3 A Simple Estimate of I ζ(s)I; 2.4 The Resulting Estimate of the Roots p; 2.5 Convergence of the Product; 2.6 Rate of Growth of the Quotient; 2.7 Rate of Growth of Even Entire Functions; 2.8 The Product Formula for ζ; Chapter 3. Riemann's Main Formula; 3.1 Introduction; 3.2 Derivation of von Mangoldt's Formula for ψ(x); 3.3 The Basic Integral Formula
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3.4 The Density of the Roots3.5 Proof of von Mangoldt's Formula for ψ (x); 3.6 Riemann's Main Formula; 3.7 Von Mangoldt's Proof of Riemann's Main Formula; 3.8 Numerical Evaluation of the Constant; Chapter 4. The Prime Number Theorem; 4.1 Introduction; 4.2 Hadamard's Proof That Re p 〈 1 for All p; 4.3 Proof That ψ(x)~x; 4.4 Proof of the Prime Number Theorem; Chapter 5. De la Vall6e Poussin's Theorem; 5.1 Introduction; 5.2 An Improvement of Re p 〈 1; 5.3 De la Vallte Poussin's Estimate of the Error; 5.4 Other Formulas for R(X); 5.5 Error Estimates and the Riemann Hypothesis
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5.6 A Postscript to de la Vallte Poussin's ProofChapter 6. Numerical Analysis of the Roots by Euler-Maclaurin Summation; 6.1 Introduction; 6.2 Euler-Maclaurin Summation; 6.3 Evaluation of II by Euler-Maclaurin Summation. Stirling's Series; 6.4 Evaluation of 3 by Euler-Maclaurin Summation; 6.5 Techniques for Locating Roots on the Line; 6.6 Techniques for Computing the Number of Roots in a Given Range; 6.7 Backlund's Estimate of N (T); 6.8 Alternative Evaluation of ζ'(0)/ζ(0); Chapter 7. The Riemann-Siege1 Formula; 7.1 Introduction; 7.2 Basic Derivation of the Formula
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9.4 Backlund's Reformulation of the Lindelöf Hypothesis
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English
Weitere Ausg.:
ISBN 0-12-232750-0
Sprache:
Englisch
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