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

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717719-3

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

MathSciNet review:
717719

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Abstract | References | Similar Articles | Additional Information

Abstract: We describe extensive computations which show that Riemann's zeta function has exactly 300,000,001 zeros of the form in the region . All these zeros are simple and lie on the line . (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 near the first observed failures of Rosser's rule.

**[1]**R. P. Brent, "On the zeros of the Riemann zeta function in the critical strip,"*Math. Comp.*, v. 33, 1979, pp. 1361-1372. MR**537983 (80g:10033)****[2]**R. P. Brent, J. van de Lune, H. J. J. te Riele & D. T. Winter, "On the zeros of the Riemann zeta function in the critical strip. II,"*Math. Comp.*, v. 39, 1982, pp. 681-688. MR**669660 (83m:10067)****[3]**E. Karkoschka & P. Werner, "Einige Ausnahmen zur Rosserschen Regel in der Theorie der Riemannschen Zetafunktion,"*Computing*, v. 27, 1981, pp. 57-69. MR**623176 (82i:10048)****[4]**J. van de Lune, H. J. J. te Riele & D. T. Winter,*Rigorous High Speed Separation of Zeros of Riemann's Zeta Function*, Report NW 113/81, October 1981, Mathematical Centre, Amsterdam.**[5]**J. van de Lune & H. J. J. te Riele,*Rigorous High Speed Separation of Zeros of Riemann's Zeta Function*, II, Report NN 26/82, June 1982, Mathematical Centre, Amsterdam.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717719-3

Keywords:
Gram blocks,
Riemann hypothesis,
Riemann zeta function,
Rosser's rule

Article copyright:
© Copyright 1983
American Mathematical Society