A. Turing and the computation of zeta zeros

"Even Alan Turing, the British mathematician who played such an important part in the British deciphering operation at Bletchley Park during the Second World War, was seduced by the fascination of the Riemann Hypothesis. In the midst of laying the theoretical foundations of what were to become digital computers, Turing designed a machine to calculate zeros of the Riemann zeta function. Turing's biographer, Andrew Hodge, writes:

'Apparently [Turing] had decided that the RH was probably false, if only because such great efforts have failed to proved it. Its falsity would mean that the zeta function did take the value zero at some point which was off the special line, in which case this point could be located by brute force, just by calculating enough values of the zeta function.'

Turing did much of his own engineering work, planning to construct a system of eighty meshing gearwheels with weights attached at specific distances from their centres. As the wheels rotated, the combined effect of different weights rotating at different distances from their axes would produce a varying turning effect - a moment, as it is called - and that moment, Turing hoped, could be estimat4ed by balancing the whole system with a counterweight. From the magnitude of this moment, after each set of rotations of the wheels, a new Riemann zero could be calculated. At least that was the theory.

Turing's friends got used to the resulting chaos of his room in King's College, Cambridge, where he had been made a Fellow in 1935:

It was liable to be found with the sort of jigsaw puzzle of gear wheels across the floor. Kenneth Harrison, now a Fellow, was invited in for a drink and found it in this state. Alan tried and lamentably failed to explain what it was all for. It was certainly far from obvious that the motion of these wheels would say anything about the regularity with which the prime numbers thinned out, in their billions of billions up to infinity. Alan made a start on doing the actual gear cutting, humping the blanks along to the engineering department in a rucksack, and spurning an offer of help from a research student. Champ [a friend, David Champernowne] lent a hand in grinding some of the wheels, which were kept in a suitcase in Alan's room."

This excerpt is from K. Sabbagh, Dr. Riemann's Zeros (Atlantic, 2002), p.89.

The machine was never finished.

More details are available here.

 


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