Alan Turing and Number Theory

Beside well-known revolutionary contributions, Alan Turing had a number of significant results in “traditional” mathematics. In particular he was very much interested in the famous Riemann Hypothesis. This hypothesis, stated by Berhard Riemann in 1859 and included by David Hilbert in his 8th problem in 1990, still remains open, being now one of the Millennium Problems. The Riemann Hypothesis predicts positions of zeros of so called zeta function, and Alan Turing developed a rigorous method for verifying the Hypothesis for the initial zeros. He also invented a machine for calculating the values of the zeta function. In contrast to celebrated imaginable Turing machines, Turing started to implement this machine but never finished because of the War.