Alan Turing and the Riemann Zeta Function

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. Most have now been superseded by later work, but one technique that he introduced is still a standard tool in the computational analysis of the zeta and related functions. It is known as Turing’s method, and keeps his name alive in those areas. Of Turing’s two published papers [27, 28] involving the Riemann zeta function ζ(s), the second is the more significant. In it, Turing reports on the first calculation of zeros of ζ(s) 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. Our primary aim in this chapter is to provide an overview of Turing’s work on the zeta function. The influence that interactions with available technology and with other researchers had on his thinking is deduced from [27, 28] as well as some unpublished manuscripts of his (available in [29]) and related correspondence, some newly discovered. (To minimize any overlap with other chapters, we do not discuss Turing’s contributions to computing