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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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

Author: Richard P. Brent
Journal: Math. Comp. 33 (1979), 1361-1372
MSC: Primary 10H05
MathSciNet review: 537983
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Abstract: We describe a computation which shows that the Riemann zeta function $ \zeta (s)$ has exactly 75,000,000 zeros of the form $ \sigma + it$ it in the region $ 0 < t < 32,585,736.4$; all these zeros are simple and lie on the line $ \sigma = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} $. (A similar result for the first 3,500,000 zeros was established by Rosser, Yohe and Schoenfeld.) Counts of the number of Gram blocks of various types and the number of failures of "Rosser's rule" are given.

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Keywords: Gram blocks, Riemann hypothesis, Riemann zeta function, Riemann-Siegel formula, Rosser's rule, Turing's theorem, zeta functions
Article copyright: © Copyright 1979 American Mathematical Society