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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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
Corrigendum: Math. Comp. 46 (1986), 771.
MathSciNet review: 717719
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1983-0717719-3
PII: S 0025-5718(1983)0717719-3
Keywords: Gram blocks, Riemann hypothesis, Riemann zeta function, Rosser's rule
Article copyright: © Copyright 1983 American Mathematical Society