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about the line x =^12 , and that as we progress vertically up and down the line x =^12 , many zeros
do lie on it: in fact, the first trillion zeros lie on this line! But do all of the non-trivial zeros lie on
the line x =^12 , or are the first trillion or so just a coincidence?
It is now generally believed that all the non-trivial zeros do lie on this line, but proving this
is one of the most celebrated unsolved challenges in the whole of mathematics. Finding just
one zero off the line would cause major havoc in number theory—and in fact throughout
mathe matics. But no-one has been able to prove the Riemann hypothesis, even after a century
and a half.
Turing and the Riemann hypothesis
In 1936, while Turing was in Princeton, the Oxford mathematician E. C. Titchmarsh^13 proved
that no zeros appear on the critical line up to a height of 1468. In the following year, working
on a suggestion of Ingham, Turing set to work seriously on the zeros of the zeta function. Like
Titchmarsh, but unlike most other mathematicians, Turing believed that the Riemann hypoth-
esis might be false, and he contemplated building a special ‘gear-wheel’ calculating machine to
search for any zeros that strayed off the critical line.
Turing wrote two papers on the Riemann hypothesis. The first, written in Cambridge in
1939 but not published until 1943, was highly technical and included numerical methods
for calculating values of the zeta function.^14 Although it refined some of Titchmarsh’s work,
Turing’s results were soon superseded, being made unnecessary by advances brought about
by the use of electronic computers. In the same year Turing submitted a proposal to the
Royal Society for the construction of a ‘zeta function machine’, with the aid of which he
hoped to extend Titchmarsh’s numerical results by a factor of about 4 (that is, up to a height
of about 6000). Due to the outbreak of the war, the construction of this machine was never
completed.
In 1948 Turing moved to the Manchester Computing Machine Laboratory, where in June
1950 he carried out his calculations on the zeros of the zeta function on the Ferranti Mark I
electronic computer. His second paper described the process in detail, as he looked for zeros
off the critical line around the height 25,000, and he also proved conclusively that none appears
up to height 1540—a ‘negligible advance’, he said, as much more had been hoped for. He
explained:^15
The calculations had been planned some time in advance, but had in fact to be carried out in
great haste. If it had not been for the fact that the computer remained in serviceable condition
for an unusually long period from 3 p.m. one afternoon to 8 a.m. the following morning it is
probable that the calculations would never have been done at all. As it was, the interval 2π·63^2
< t < 2π·64^2 [24,938 to 25,735] was investigated during that period, and very little more was
accomplished . . . The interval 1414 < t < 1608 was investigated and checked, but unfortunately
at this point the machine broke down and no further work was done. Furthermore this interval
was subsequently found to have been run with a wrong error value, and the most that can
consequently be asserted with certainty is that the zeros lie on the critical line up to t = 1540,
Titchmarsh having investigated as far as 1468.
Further information about Turing and the Riemann hypothesis can be found in the articles
by Booker^16 and Hejhal and Odlyzko.^17