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the zeta zero approximator

A blueprint
The Turing Digital Archive contains a single tantalizing blueprint image for an elaborate gear-driven mechanical calculator that Turing proposed to build in 1939, which would have helped to make progress in verifying the Riemann Hypothesis. It was to be a very special-purpose device for adding up sine components in the various ratios needed to perform calculations using the Riemann-Siegel theta function, which was a new development in the 30’s.

Table of values from blueprint
The table in the blueprint contains the ratios he would need for mechanical linkage, although as Bill Casselman points out, the table actually contains several calculation errors which would have eventually caused some problems. [My personal favorite is the column for ratios in log base 8.]

From Turing’s application to the Royal Society: “It is proposed to make calculations of the Riemann zeta-function on the critical line for 1,450 < t < 6,000 with a view to discovering whether all the zeros of the function in this range of t lie on the critical line. An investigation for 0 < t < 1,464 has already been made by Titchmarsh. The most laborious part of such calculations consists in the evaluation of certain trigonometrical sums
An intermediate equation
In the present calculation it is intended to evaluate these sums approximately in most cases by the use of apparatus somewhat similar to what is used for tide prediction. When this method does not give sufficient accuracy it will be necessary to revert to the straightforward calculation of the trigonometric sums, but this should be only rarely necessary. I am hoping that the use of the tide-predicting machine will reduce the amount of such calculation necessary in a ratio of 50:1 or better. It will not be feasible to use already existing tide predictors because the frequencies occurring in the tide problems are entirely different from those occurring in the zeta function problem. I shall be working in collaboration with D. C. MacPhail, a research student who is an engineer. We propose to do most of the machineshop work ourselves, and are therefore applying only for the cost of materials, and some preliminary computation.”

Although this physical machine was never finished, due to the arrival of World War II, Turing continued to putter with the Riemann Hypothesis throughout his career via the zeta function, eventually becoming the first person to use an electronic computer to calculate zeroes, and thereby extending the upper limit for known zeroes to t < 1540. [Minuscule by today’s standards, but not bad for work done with paper tape in raw base 32 on a machine with a little over 25,000 bits of memory!] He also devised what is now called “Turing’s method” for easier computational analysis of the function. These exploits are detailed in his papers “A method for the calculation of the zeta-function” and “Some calculations of the Riemann zeta-function,” which are both widely referenced in contemporary math papers.