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The Riemann hypothesis : a resource for the afficionado and virtuoso alike / P. Borwein ... [et al.].

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Language:
English.
Publication date:
2008
Imprint:
New York ; London : Springer, 2008.
Format:
  • Book
  • xiv., 533 p. : ill. ; 24 cm.
Bibliography:
Includes bibliographical references and index.
Contents:
  • pt. 1. Introduction to the Riemann hypothesis
  • 1. Why this book
  • 1.1. The Holy Grail
  • 1.2. Riemann's zeta and Liousville's lambda
  • 1.3. The prime number theorem
  • 2. Analytic preliminaries
  • 2.1. The Riemann zeta function
  • 2.2. Zero-free region
  • 2.3. Counting the zeros of [cedilla](s)
  • 2.4. Hardy's theorem
  • 3. Algorithms for calculating [cedilla](s)
  • 3.1. Euler-MacLaurin summation
  • 3.2. Backlund
  • 3.3. Hardy's function
  • 3.4. The Riemann-Siegel formula
  • 3.5. Gram's law
  • 3.6. Turing
  • 3.7. The Odlyzko-Schönhage algorithm
  • 3.8. A simple algorithm for the zeta function
  • 3.9. Further reading
  • 4. Empirical evidence
  • 4.1. Verification in an interval
  • 4.2. A brief history of computational evidence
  • 4.3. The Riemann hypothesis and random matrices
  • 4.4. The Skewes number.
  • 5. Equivalent statements
  • 5.1. Number-theoretic equivalences
  • 5.2. Analytic equivalences
  • 5.3. Other equivalences
  • 6. Extensions of the Riemann hypothesis
  • 6.1. The Riemann hypothesis
  • 6.2. The generalized Riemann hypothesis
  • 6.3. The extended Riemann hypothesis
  • 6.4. An equivalent extended Riemann hypothesis
  • 6.5. Another extended Riemann hypothesis
  • 6.6. The Grand Riemann hypothesis
  • 7. Assuming the Riemann hypothesis and its extensions
  • 7.1. Another proof of the prime number theorem
  • 7.2. Goldbach's conjecture
  • 7.3. More Goldbach
  • 7.4. Primes in a given interval
  • 7.5. The least prime in arithmetic progressions
  • 7.6. Primality testing
  • 7.7. Artin's primitive root conjecture
  • 7.8. Bounds on Dirichlet L-series
  • 7.9. The Lindelöf hypothesis
  • 7.10. Titchmarsh's [delta](T) function
  • 7.11. Mean values of [cedilla](s).
  • 8. Failed attempts at proof
  • 8.1. Stieltjes and Mertens' conjecture
  • 8.2. Hans Rademacher and false hopes
  • 8.3. Turán's condition
  • 8.4. Louis de Branges's approach
  • 8.5. No really good idea
  • 9. Formulas
  • 10. Timeline
  • pt. 2. Original papers
  • 11. Expert witnesses
  • 11. 1. E. Bombieri (2000-2001)
  • 11.2. P. Sarnak (2004)
  • 11.3. J.B. Conrey (2003)
  • 11.4. A. Ivić (2003)
  • 12. The experts speak for themselves
  • 12.1. P.L. Chebyshev (1852)
  • 12.2. B. Riemann (1859)
  • 12.3. J. Hadamard (1896)
  • 12.4. C. de la Vallée Poussin (1899)
  • 12.5. G.H. Hardy (1914)
  • 12.6. G.H. Hardy (1915)
  • 12.7. G.H. Hardy and J.E. Littlewood (1915)
  • 12.8. A. Weil (1941)
  • 12.9. P. Turán (1948)
  • 12.10. A. Selberg (1949)
  • 12.11. P. Erdoʺs (1949)
  • 12.12. S. Skewes (1955)
  • 12.13. C.B. Haselgrove (1958)
  • 12.14. H. Montgomery (1973)
  • 12.15. D.J. Newman (1980)
  • 12.16. J. Korevaar (1982)
  • 12.17. H. Daboussi (1984)
  • 12.18. A. Hildebrand (1986)
  • 12.19. D. Goldston and H. Montgomery (1987)
  • 12.20. M. Agrawal, N. Kayal, and N. Saxena (2004)
  • References
  • References
  • Index.
Contributor:
Borwein, Peter B.
Series:
CMS books in mathematics.
Subjects:
ISBN:
9780387721255
0387721258
0387721266
9780387721262
Publisher No.:
11609056

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