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

van de Lune, J.; te Riele, H. J. J.

doi:10.1090/S0025-5718-1983-0717719-3

On the zeros of the Riemann zeta function in the critical strip. III
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by J. van de Lune and H. J. J. te Riele PDF

Math. Comp. 41 (1983), 759-767 Request permission

Corrigendum: Math. Comp. 46 (1986), 771.

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

Richard P. Brent, On the zeros of the Riemann zeta function in the critical strip, Math. Comp. 33 (1979), no. 148, 1361–1372. MR 537983, DOI 10.1090/S0025-5718-1979-0537983-2
R. P. Brent, J. van de Lune, H. J. J. te Riele, and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. II, Math. Comp. 39 (1982), no. 160, 681–688. MR 669660, DOI 10.1090/S0025-5718-1982-0669660-1
E. Karkoschka and P. Werner, Einige Ausnahmen zur Rosserschen Regel in der Theorie der Riemannschen Zetafunktion, Computing 27 (1981), no. 1, 57–69 (German, with English summary). MR 623176, DOI 10.1007/BF02243438

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