## ‘A Method for the Calculation of the Zeta-Function’, pp. 180-197 in Proceedings of the London Mathematical Society, Series 2, Vol. 48, No. 3, December 15, 1943. London: C. F. Hodgson and Son, 1943. [Offered with:] ‘Some calculations of the Riemann zeta-function,’ pp. 99-117 in ibid., Series 3, Vol. 3, No. 9, March 1953.

London: C. F. Hodgson and Son, 1943; 1953.

First edition, journal issues in the original printed wrappers, of Turing’s ground-breaking work outlining a method to decide the most famous open problem in mathematics, the so-called Riemann hypothesis. This is a conjecture about the location of the zeros of the ‘Riemann zeta function’ – it asserts that, apart from some ‘trivial’ zeros, they all lie on a certain ‘critical line.’ If true, this would have enormous implications for the study of prime numbers. Turing had worked on the zeta function since 1939 and in ‘A Method for the Calculation of the Zeta-Function’ he outlined a method of calculating the zeros using a mechanical computer. “The Turing archive contains a sketch of a proposal, in 1939, to build an analog computer that would calculate approximate values for the Riemann zeta-function on the critical line. His ingenious method was published in 1943 [as the present work]” (Downey, p. 11). Although he received a grant to build his zeta-function machine, the outbreak of World War II, and Turing’s role in it as cryptanalyst, postponed the work, and the machine was never constructed. After the War, Turing returned to the Riemann hypothesis and developed a new procedure, now known as ‘Turing’s method’, for checking the Riemann Hypothesis (described in Section 4 of the 1953 paper). He then used the Manchester Mark I digital computer to implement this method. “Of Turing’s two published papers [both offered here] on the Riemann zeta function, the second is the more significant. In that paper, Turing reports on the first calculation of zeros of [the zeta function] ever done with the aid of an electronic digital computer. It was in developing the theoretical underpinnings for this work that Turing’s method first came into existence” (Hedjhal & Odlyzko, p. 265). ‘Some calculations of the Riemann zeta-function’ was Turing’s last published mathematical paper. “This work was one of the first announcing a new chapter in which experimental mathematics performed with computers would play an important role” (Mezzadri and Snaith, *Recent Perspectives in Random Matrix and Number Theory*). Rare on the market in unrestored original printed wrappers – we know of only one copy of the first paper at auction, in the Weinreb Computer Collection (Bloomsbury Book Auctions, 28 October 1999), and no other copy of the second.

The Riemann zeta function is defined as the sum of an infinite series

ζ(*s*) = 1/1* ^{s}* + 1/2

*+ 1/3*

^{s}*+ 1/4*

^{s}*+ …..*

^{s}This actually makes sense when

*s*is any

*complex*number (except

*s*= 1, when the sum is infinite). It is known that ζ(

*s*) = 0 when

*s*= –2, –4, –6, … – these are called the ‘trivial zeros’. The Riemann hypothesis (RH) is the assertion that all the non-trivial zeros are complex numbers of the form

*s*= ½ +

*t*√-1, where

*t*is a real number – these complex numbers form a line in the complex plane, called the ‘critical line’. The RH, first put forward by Bernhard Riemann in 1859, is known to be true for the first 10

^{13}non-trivial zeros, but remains unproven. “The RH is widely regarded as the most famous unsolved problem in mathematics. It was one of the 23 famous problems selected by [David] Hilbert in 1900 as among the most important in mathematics, and it is one of the seven Millennium Problems selected by the Clay Mathematics Institute in 2000 as the most important for the 21st century” (Hedjhal & Odlyzko, p. 266).

“The first computations of zeros of the zeta function were performed by Riemann, and likely played an important role in his posing of the RH as a result likely to be true. His computations were carried out by hand, using an advanced method that is known today as the Riemann-Siegel formula. Both the method and Riemann’s computations that utilized it remained unknown to the world-at-large until the early 1930s, when they were found in Riemann’s unpublished papers by C. L. Siegel … In the mid-1930s, after Siegel’s publication of the Riemann-Siegel formula, [the Oxford mathematician] E. C. Titchmarsh obtained a grant for a larger computation. With the assistance of L. J. Comrie, tabulating machines, some ‘computers’ (as the mostly female operators of such machinery were called in those days), and the recently published algorithm, Titchmarsh established that the 1041 nontrivial zero with 0 < *t* < 1468 all satisfied the RH” (*ibid*., p. 268).

“Turing encountered the Riemann zeta function as a student, and developed a life-long fascination with it. Though his research in this area was not a major thrust of his career, he did make a number of pioneering contributions” (*ibid*., p. 266). “Apparently he had decided that the Riemann hypothesis was probably false, if only because such great efforts had failed to prove 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 … There were two aspects to the problem. Riemann’s zeta-function was defined as the sum of an infinite number of terms, and although this sum could be re-expressed in many different ways, any attempt to evaluate it would in some way involve making an approximation. It was for the mathematician to find a good approximation, and to prove that it was good: that the error involved was sufficiently small. Such work did not involve computation with numbers, but required highly technical work with the calculus of complex numbers. Titchmarsh had employed a certain approximation which – rather romantically – had been exhumed from Riemann’s own papers at Göttingen where it had lain for seventy years. But for extending the calculation to thousands of new zeroes a fresh approximation was required; and this Alan set out to find and to justify.

“The second problem, quite different, was the ‘dull and elementary’ one of actually doing the computation, with numbers substituted into the approximate formula, and worked out for thousands of different entries. It so happened that the formula was rather like those which occurred in plotting the positions of the planets, because it was of the form of a sum of circular functions with different frequencies. It was for this reason that Titchmarsh had contrived to have the dull repetitive work of addition, multiplication, and of looking up of entries in cosine tables done by the same punched-card methods that were used in planetary astronomy. But it occurred to Alan that the problem was very similar to another kind of computation which was also done on a large practical scale – that of tide prediction. Tides could also be regarded as the sum of a number of waves of different periods: daily, monthly, yearly oscillations of rise and fall. At Liverpool there was a machine which performed the summation automatically, by generating circular motions of the right frequencies and adding them up. It was a simple analogue machine; that is, it created a physical analogue of the mathematical function that had to be calculated. This was a quite different idea from that of the Turing machine, which would work away on a finite, discrete, set of symbols. This tide-predicting machine, like a slide rule, depended not on symbols, but on the measurement of lengths. Such a machine, Alan had realised, could be used on the zeta-function calculation, to save the dreary work of adding, multiplying, and looking up of cosines.

“Alan must have described this idea to Titchmarsh, for a letter from him dated 1 December 1937 approved of this programme of extending the calculation, and mentioned: ‘I have seen the tide-predicting machine at Liverpool, but it did not occur to me to use it in this way’ (Hodges, pp. 140-2).

“On 24 March [1939] he applied to the Royal Society for a grant to cover the cost of constructing it, and on their questionnaire wrote, ‘Apparatus would be of little permanent value. It could be added to for the purpose of carrying out similar calculations for a wider range of *t *and might be used for some other investigations connected with the zeta-function. I cannot think of any applications that would not be connected with the zeta-function.’ Hardy and Titchmarsh were quoted as referees for the application, which won the requested £40. The idea was that although the machine could not perform the required calculation exactly, it could locate the places where the zeta-function took a value near zero, which could then be tackled by a more exact hand computation. Alan reckoned it would reduce the amount of work by a factor of fifty. Perhaps as important, it would be a good deal more fun” (Hodges, pp. 155-6).

Turing started to work on the construction of his zeta-function machine, but the work was interrupted by the outbreak of World War II, and the machine was never built. “We do not know how well Turing's zeta function machine would have worked, had it been built. At least one special zeta function computer was constructed to a different design later by van der Pol (1947). By that time, though, electronic digital computers were becoming available, and Turing was the first one to utilise them to investigate the zeta function” (Hedjhal & Odlyzko, p. 268).

Soon after his involvement in the war effort ended, Turing set about plans for a general-purpose digital computer. He submitted a detailed design for the Automatic Computing Engine (ACE) to the National Physical Laboratory in early 1946. Turing’s design drew on both his theoretical work ‘On Computable Numbers’ from a decade earlier, and the practical knowledge gained during the war from working at Bletchley Park, where the Colossus machines were developed and used. But there were several delays in realizing Turing’s plans. The existence and capabilities of the Colossus machines were classified top-secret for decades after the

war, so Turing was forbidden from disclosing what he already knew to be achievable. Even a scaled-down plan for a Pilot ACE met with so many bureaucratic delays that Turing resigned his post at the NPL and moved in late 1948 to Manchester at the invitation of his former lecturer at Cambridge, Max Newman. The Manchester Mark I was operational a few months after Turing’s arrival, and Newman and Turing looked for mathematical problems the new computer could help to solve.

“In 1950, he used the Manchester Mark I Electronic Computer to extend the Titchmarsh verification of the RH to the first 1104 zeros of the zeta function, the ones with 0 < *t *< 1540. This was a very small extension, but it represented a triumph of perseverance over a promising new technology that was still suffering from teething problems. In Turing’s words: ‘[I]f it had not been for the fact that the computer remained in serviceable condition for an unusually long period from 3pm one afternoon to 8am the following morning it is probable that the calculations would never have been done at all.’ These days, when even our simple consumer devices have gigabytes of memory, it is instructive to recall that the machine available to Turing had a grand total of 25,600 bits of memory and that Turing worked directly output ‘punched on a teleprint tape’ in base 32. That Turing stayed up all through the night conveys some idea of how interesting he found this experiment” (*ibid*.).

The first 10 trillion zeros of the zeta function have been found to obey the RH, as has the 10^{32}nd zero and hundreds of its neighbours; all such calculations continue to rely on Turing’s method as an essential ingredient. In recent decades computers have come to play an increasingly important role in other mathematical proofs: one has only to note the computer-assisted proofs of the ‘Four colour theorem’ and ‘Kepler’s conjecture’. Such proofs have proved controversial, but Turing’s view was expressed clearly in the 1953 paper: ‘If definite rules are laid down as to how the computation is to be done one can predict bounds for the errors throughout. When the computations are done by hand there are serious practical difficulties about this. The [human] computer will probably have his own ideas as to how certain steps should be done … However, if the calculations are being done by an automatic computer one can feel sure that this kind of indiscipline does not occur.’

Downey (ed.), *Turing’s Legacy*: *Developments from Turing’s Ideas in Logic* (2014); Hedjhal & Odlyzko, ‘Alan Turing and the Riemann zeta function,’ pp. 265-279 in Cooper & van Leeuwen (eds.), *Alan Turing*: *His Work and Impact* (2013); Hodges, *Alan Turing*: *the Enigma* (1983).

Two vols., large 8vo, pp. 161-240 & 1-128. Original printed wrappers (first volume with tiny chip from upper right corner of front wrapper, edges of wrappers lightly browned, second volume with small closed tear at top of spine). Very good copies.

Item #4646

**
Price:
$3,850.00
**