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On the Zeros of the Riemann Zeta Function in the Critical Strip. IV

J. van de Lune, H. J. J. te Riele and D. T. Winter
Mathematics of Computation
Vol. 46, No. 174 (Apr., 1986), pp. 667-681
DOI: 10.2307/2008005
Stable URL: http://www.jstor.org/stable/2008005
Page Count: 15
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On the Zeros of the Riemann Zeta Function in the Critical Strip. IV
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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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