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

47. 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.

Abstract: dvi (3K), pdf (80K), ps (28K).

Paper: pdf (1012K).


We describe a computation which shows that the Riemann zeta function has exactly 75,000,000 zeros of the form sigma + i.t in the region 0 <t <32,585,736.4; all these zeros are simple and lie on the line sigma = 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.


1. Page 1361, line -9 (not counting footnotes): replace H(10) by H(15).

2. Page 1362, equation (2.3): replace B2k by |B2k|.

3. Page 1362, two lines after equation (2.3): the bound on |rn(t)| is incorrect.
(The error is only significant for small t and did not affect the computation of zeros of the zeta function.)


For an extension of the computation to the first 2 × 108 zeros see Brent, van de Lune, te Riele and Winter [7081].

Wedeniwski's ZetaGrid project, using essentially the same algorithm on more than 600 computers, has pushed the bound past 1013 zeros.

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