## The first 200,000,001 zeros of Riemann's zeta function

81. 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, Jr. and R. Tijdeman),
Mathematical Centre Tracts 154,
Mathematisch Centrum, Amsterdam, 1982, 389-403.
MR 84h:10003, 84d:10004.
Abstract:
dvi (3K),
pdf (81K),
ps (28K).

Paper:
pdf (1315K).

## Abstract

We describe extensive computations which show that Riemann's zeta function
has exactly 200,000,001 zeros of the form
*sigma* + i.*t*
in the region 0 <*t* < 81,702,130.19;
all these zeros are simple and lie on the line *sigma* = 1/2.
This extends a result for the first 81,000,001
zeros, established by Brent in
[47].
Counts of the numbers of Gram blocks of various types
and the failures of "Rosser's rule" are given.
## Comments

This is a more detailed version of
[70].
A note "added in proof" (December 1982) states that the result has
been extended to the first 307,000,000 zeros.
This was sufficient to settle
a bet between Enrico Bombieri and Don Zagier -
see Marcus du Sautoy,
* The Music of the Primes*, Harper-Collins, 2003.

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