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On the zeros of the Riemann zeta function in the critical strip. III


Authors: J. van de Lune and H. J. J. te Riele
Journal: Math. Comp. 41 (1983), 759-767
MSC: Primary 11M26; Secondary 11-04, 11Y35
DOI: https://doi.org/10.1090/S0025-5718-1983-0717719-3
Corrigendum: Math. Comp. 46 (1986), 771.
MathSciNet review: 717719
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Abstract | References | Similar Articles | Additional Information

Abstract: We describe extensive computations which show that Riemann's zeta function $ \zeta (s)$ has exactly 300,000,001 zeros of the form $ \sigma + it$ in the region $ 0 < t < 119,590,809.282$. All these zeros are simple and lie on the line $ \sigma = \frac{1}{2}$. (This extends a similar result for the first 200,000,001 zeros, established by Brent, van de Lune, te Riele and Winter in Math. Comp., v. 39, 1982, pp. 681-688.) Counts of the numbers of Gram blocks of various types and the failures of "Rosser's rule" are given, together with some graphs of the function $ Z(t)$ near the first observed failures of Rosser's rule.


References [Enhancements On Off] (What's this?)

  • [1] R. P. Brent, "On the zeros of the Riemann zeta function in the critical strip," Math. Comp., v. 33, 1979, pp. 1361-1372. MR 537983 (80g:10033)
  • [2] R. P. Brent, J. van de Lune, H. J. J. te Riele & D. T. Winter, "On the zeros of the Riemann zeta function in the critical strip. II," Math. Comp., v. 39, 1982, pp. 681-688. MR 669660 (83m:10067)
  • [3] E. Karkoschka & P. Werner, "Einige Ausnahmen zur Rosserschen Regel in der Theorie der Riemannschen Zetafunktion," Computing, v. 27, 1981, pp. 57-69. MR 623176 (82i:10048)
  • [4] J. van de Lune, H. J. J. te Riele & D. T. Winter, Rigorous High Speed Separation of Zeros of Riemann's Zeta Function, Report NW 113/81, October 1981, Mathematical Centre, Amsterdam.
  • [5] J. van de Lune & H. J. J. te Riele, Rigorous High Speed Separation of Zeros of Riemann's Zeta Function, II, Report NN 26/82, June 1982, Mathematical Centre, Amsterdam.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1983-0717719-3
Keywords: Gram blocks, Riemann hypothesis, Riemann zeta function, Rosser's rule
Article copyright: © Copyright 1983 American Mathematical Society

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