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Publisher:

Dover Publications

Publication Date:

2001

Number of Pages:

330

Format:

Paperback

Price:

14.95

ISBN:

0486417409

Category:

Monograph

We do not plan to review this book.

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 zeta (s) | |||||||

1.5 | Values of zeta (s) | |||||||

1.6 | First Proof of the Functional Equation | |||||||

1.7 | Second Proof of the Functional Equation | |||||||

1.8 | The Function xi (s) | |||||||

1.9 | The Roots rho of xi | |||||||

1.10 | The Product Representation of xi (s) | |||||||

1.11 | The Connection between zeta (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) | |||||||

1.15 | The Term Involving the Roots rho | |||||||

1.16 | The Remaining Terms | |||||||

1.17 | The Formula for pi (x) | |||||||

1.18 | The Density dJ | |||||||

1.19 | Questions Unresolved by Riemann | |||||||

Chapter 2. | The Product Formula for xi | |||||||

2.1 | Introduction | |||||||

2.2 | Jensen's Theorem | |||||||

2.3 | A Simple Estimate of absolute value of |xi (s)| | |||||||

2.4 | The Resulting Estimate of the Roots rho | |||||||

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 xi | |||||||

Chapter 3. | Riemann's Main Formula | |||||||

3.1 | Introduction | |||||||

3.2 | Derivation of von Mangoldt's formula for psi (x) | |||||||

3.3 | The Basic Integral Formula | |||||||

3.4 | The Density of the Roots | |||||||

3.5 | Proof of von Mangoldt's Formula for psi (x) | |||||||

3.6 | Riemann's Main Formula | |||||||

3.7 | Von Mangoldt's Proof of Reimann'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 rho<1 for All rho | |||||||

4.3 | Proof That psi (x) ~ x | |||||||

4.4 | Proof of the Prime Number Theorem | |||||||

Chapter 5. | De la Vallée Poussin's Theorem | |||||||

5.1 | Introduction | |||||||

5.2 | An Improvement of Re rho<1 | |||||||

5.3 | De la Vallée Poussin's Estimate of the Error | |||||||

5.4 | Other Formulas for pi (x) | |||||||

5.5 | Error Estimates and the Riemann Hypothesis | |||||||

5.6 | A Postscript to de la Vallée Poussin's Proof | |||||||

Chapter 6. | Numerical Analysis of the Roots by Euler-Maclaurin Summation | |||||||

6.1 | Introduction | |||||||

6.2 | Euler-Maclaurin Summation | |||||||

6.3 | Evaluation of PI by Euler-Maclaurin Summation. Stirling's Series | |||||||

6.4 | Evaluation of zeta 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 zeta'(0)/zeta(0) | |||||||

Chapter 7. | The Riemann-Siegel Formula | |||||||

7.1 | Introduction | |||||||

7.2 | Basic Derivation of the Formula | |||||||

7.3 | Estimation of the Integral away from the Saddle Point | |||||||

7.4 | First Approximation to the Main Integral | |||||||

7.5 | Higher Order Approximations | |||||||

7.6 | Sample Computations | |||||||

7.7 | Error Estimates | |||||||

7.8 | Speculations on the Genesis of the Riemann Hypothesis | |||||||

7.9 | The Riemann-Siegel Integral Formula | |||||||

Chapter 8. | Large-Scale Computations | |||||||

8.1 | Introduction | |||||||

8.2 | Turing's Method | |||||||

8.3 | Lehmer's Phenomenon | |||||||

8.4 | Computations of Rosser, Yohe, and Schoenfeld | |||||||

Chapter 9. | The Growth of Zeta as t --> infinity and the Location of Its Zeros | |||||||

9.1 | Introduction | |||||||

9.2 | Lindelöf's Estimates and His Hypothesis | |||||||

9.3 | The Three Circles Theorem | |||||||

9.4 | Backlund's Reformulation of the Lindelöf Hypothesis | |||||||

9.5 | The Average Value of S(t) Is Zero | |||||||

9.6 | The Bohr-Landau Theorem | |||||||

9.7 | The Average of absolute value |zeta(s)| superscript 2 | |||||||

9.8 | Further Results. Landau's Notation o, O | |||||||

Chapter 10. | Fourier Analysis | |||||||

10.1 | Invariant Operators on R superscript + and Their Transforms | |||||||

10.2 | Adjoints and Their Transforms | |||||||

10.3 | A Self-Adjoint Operator with Transform xi (s) | |||||||

10.4 | The Functional Equation | |||||||

10.5 | 2 xi (s)/s(s - 1) as a Transform | |||||||

10.6 | Fourier Inversion | |||||||

10.7 | Parseval's Equation | |||||||

10.8 | The Values of zeta (-n) | |||||||

10.9 | Möbius Inversion | |||||||

10.10 | Ramanujan's Formula | |||||||

Chapter 11. | Zeros on the Line | |||||||

11.1 | Hardy's Theorem | |||||||

11.2 | There Are at Least KT Zeros on the Line | |||||||

11.3 | There Are at Least KT log T Zeros on the Line | |||||||

11.4 | Proof of a Lemma | |||||||

Chapter 12. | Miscellany | |||||||

12.1 | The Riemann Hypothesis and the Growth of M(x) | |||||||

12.2 | The Riemann Hypothesis and Farey Series | |||||||

12.3 | Denjoy's Probabilistic Interpretation of the Riemann Hypothesis | |||||||

12.4 | An Interesting False Conjecture | |||||||

12.5 | Transforms with Zeros on the Line | |||||||

12.6 | Alternative Proof of the Integral Formula | |||||||

12.7 | Tauberian Theorems | |||||||

12.8 | Chebyshev's Identity | |||||||

12.9 | Selberg's Inequality | |||||||

12.10 | Elementary Proof of the Prime Number Theorem | |||||||

12.11 | Other Zeta Functions. Weil's Theorem | |||||||

Appendix. | On the Number of Primes Less Than a Given Magnitude (By Bernhard Riemann) | |||||||

References; Index | ||||||||

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