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| Named Person: | Bernhard Riemann; Bernhard Riemann |
|---|---|
| Material Type: | Internet resource |
| Document Type: | Book, Internet Resource |
| All Authors / Contributors: |
Peter B Borwein; et al |
| ISBN: | 9780387721255 0387721258 |
| OCLC Number: | 190760103 |
| Description: | xiv.,533 p. : ill. ; 24 cm. |
| 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. |
| Series Title: | CMS books in mathematics |
| Responsibility: | P. Borwein ... [et al.]. |
| More information: |
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