ON THE ZEROS OF THE RIEMANN ZETA FUNCTION IN THE CRITICAL STRIP

RICHARD P. BRENT

Abstract

We describe a computation which shows that the Riemann zeta function ζ(s) has exactly 75,000,000 zeros of the form σ + it in the region 0 < t < 32,585,736.4; all these zeros are simple and lie on the line σ = 1/2. (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.

Comments

Only the Abstract is given here. The full paper appeared as [1]. For further work, see [2, 3].

References

1. R. P. Brent, “On the zeros of the Riemann zeta function in the critical strip”, Mathematics of Computation 33 (1979), 1361–1372. MR 80g:10033, Zbl 422.10031. A preliminary version appeared as Computation of the Zeros of the Riemann Zeta Function in the Critical Strip, Report TR CMU-CS-78-148, Department of Computer Science, Carnegie-Mellon University (November 1978), 27 pp. A progress report appeared as “The first 40,000,000 zeros of the Riemann zeta function lie on the critical line”, AMS Notices 24 (1977), A–417, 77T–A161. See also RMT 11, Mathematics of Computation 31 (1977), 803–804. rpb047.

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”, Mathematics of Computation 39 (1982), 681–688. MR 83m:10067. Corrigendum ibid 46 (1986), 771. MR 87e:11103. Also appeared as Technical Report NW 121/82, Mathematisch Centrum, Amsterdam; and as Report TR-CS-82-01, Department of Computer Science, ANU, January 1982, 13 pp. rpb070.

3. R. P. Brent, J. van de Lune, H. J. J. te Riele and D. T. Winter, “The first 200,000,001 zeros of Riemann’s zeta function”, in Computational Methods in Number Theory (edited by H. W. Lenstra and R. Tydeman), Math. Centrum, Amsterdam, 1982, 389–403. MR 84h:10003, 84d:10004. rpb081.

Department of Computer Science, Australian National University, Canberra

1991 Mathematics Subject Classification. Primary 11M26; Secondary 11-04, 30-04, 65E05. Key words and phrases. Gram blocks, Riemann hypothesis, Riemann zeta function, Riemann-Siegel

formula, Rosser’s rule, Turing’s theorem, zeta functions.

The author thanks Enrico Bombieri, John Coates, Harold Edwards, R. Sherman Lehman, Derrick H. Lehmer, Hugh Montgomery, Lowell Schoenfeld and Daniel Shanks for many useful suggestions, and the Australian National University for the provision of computer time.

Received October 4, 1978; revised February 8, 1979. Copyright \begin{matrix}c \\ ◯ \end{matrix} 1979, American Mathematical Society. Comments \begin{matrix}c \\ ◯ \end{matrix} 1993, R. P. Brent. rpb047a typeset using AMS-L^{A}TEX