Riemann's Zeta FunctionPaperback Dover Books on Mathematics
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- Publisher: Dover Publications Inc.
- Format: Paperback | 330 pages
- Dimensions: 109mm x 183mm x 18mm | 204g
- Publication date: 28 March 2003
- Publication City/Country: New York
- ISBN 10: 0486417409
- ISBN 13: 9780486417400
- Edition statement: Dover.
- Illustrations note: 3 figures
- Sales rank: 156,688
Superb high-level study of one of the most influential classics in mathematics examines landmark 1859 publication entitled "On the Number of Primes Less Than a Given Magnitude," and traces developments in theory inspired by it. Topics include Riemann's main formula, the prime number theorem, the Riemann-Siegel formula, large-scale computations, Fourier analysis, and other related topics. English translation of Riemann's original document appears in the Appendix.
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Table of contents
Preface; AcknowledgmentsChapter 1. Riemann's Paper1.1 The Historical Context of the Paper1.2 The Euler Product Formula1.3 The Factorial Function1.4 The Function zeta (s)1.5 Values of zeta (s)1.6 First Proof of the Functional Equation1.7 Second Proof of the Functional Equation1.8 The Function xi (s)1.9 The Roots rho of xi1.10 The Product Representation of xi (s)1.11 The Connection between zeta (s) and Primes1.12 Fourier Inversion1.13 Method for Deriving the Formula for J(x)1.14 The Principal Term of J(x)1.15 The Term Involving the Roots rho1.16 The Remaining Terms1.17 The Formula for pi (x)1.18 The Density dJ1.19 Questions Unresolved by RiemannChapter 2. The Product Formula for xi2.1 Introduction2.2 Jensen's Theorem2.3 A Simple Estimate of absolute value of |xi (s)|2.4 The Resulting Estimate of the Roots rho2.5 Convergence of the Product2.6 Rate of Growth of the Quotient2.7 Rate of Growth of Even Entire Functions2.8 The Product Formula for xiChapter 3. Riemann's Main Formula3.1 Introduction3.2 Derivation of von Mangoldt's formula for psi (x)3.3 The Basic Integral Formula3.4 The Density of the Roots3.5 Proof of von Mangoldt's Formula for psi (x)3.6 Riemann's Main Formula3.7 Von Mangoldt's Proof of Reimann's Main Formula3.8 Numerical Evaluation of the ConstantChapter 4. The Prime Number Theorem4.1 Introduction4.2 Hadamard's Proof That Re rho infinity and the Location of Its Zeros9.1 Introduction9.2 Lindelöf's Estimates and His Hypothesis9.3 The Three Circles Theorem9.4 Backlund's Reformulation of the Lindelöf Hypothesis9.5 The Average Value of S(t) Is Zero9.6 The Bohr-Landau Theorem9.7 The Average of absolute value |zeta(s)| superscript 29.8 Further Results. Landau's Notation o, OChapter 10. Fourier Analysis10.1 Invariant Operators on R superscript + and Their Transforms10.2 Adjoints and Their Transforms10.3 A Self-Adjoint Operator with Transform xi (s)10.4 The Functional Equation10.5 2 xi (s)/s(s - 1) as a Transform10.6 Fourier Inversion10.7 Parseval's Equation10.8 The Values of zeta (-n)10.9 Möbius Inversion10.10 Ramanujan's FormulaChapter 11. Zeros on the Line11.1 Hardy's Theorem11.2 There Are at Least KT Zeros on the Line11.3 There Are at Least KT log T Zeros on the Line11.4 Proof of a LemmaChapter 12. Miscellany12.1 The Riemann Hypothesis and the Growth of M(x)12.2 The Riemann Hypothesis and Farey Series12.3 Denjoy's Probabilistic Interpretation of the Riemann Hypothesis12.4 An Interesting False Conjecture12.5 Transforms with Zeros on the Line12.6 Alternative Proof of the Integral Formula12.7 Tauberian Theorems12.8 Chebyshev's Identity12.9 Selberg's Inequality12.10 Elementary Proof of the Prime Number Theorem12.11 Other Zeta Functions. Weil's TheoremAppendix. On the Number of Primes Less Than a Given Magnitude (By Bernhard Riemann)References; Index