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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. IV


Authors: J. van de Lune, H. J. J. te Riele and D. T. Winter
Journal: Math. Comp. 46 (1986), 667-681
MSC: Primary 11M26; Secondary 11-04, 11Y35, 30C15
MathSciNet review: 829637
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Abstract: Very extensive computations are reported which extend and, partly, check previous computations concerning the location of the complex zeros of the Riemann zeta function. The results imply the truth of the Riemann hypothesis for the first 1,500,000,001 zeros of the form $ \sigma + it$ in the critical strip with $ 0 < t < 545,439,823.215$, i.e., all these zeros have real part $ \sigma = 1/2$. Moreover, all these zeros are simple. Various tables are given with statistical data concerning the numbers and first occurrences of Gram blocks of various types; the numbers of Gram intervals containing m zeros, for $ m = 0,1,2,3$ and 4; and the numbers of exceptions to "Rosser's rule" of various types (including some formerly unobserved types). Graphs of the function $ Z(t)$ are given near five rarely occurring exceptions to Rosser's rule, near the first Gram block of length 9, near the closest observed pair of zeros of the Riemann zeta function, and near the largest (positive and negative) found values of $ Z(t)$ at Gram points. Finally, a number of references are given to various number-theoretical implications.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1986-0829637-3
PII: S 0025-5718(1986)0829637-3
Keywords: Riemann hypothesis, Riemann zeta function, Gram blocks, Rosser's rule
Article copyright: © Copyright 1986 American Mathematical Society



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