Hilbert’s Eighth and Tenth Problems

The notion of listable set is very broad and can be found in a surprising variety of contexts. Here is one such example.

Hilbert included into his 8th problem an outstanding conjecture, the famous Rie- mann’s hypothesis. In its original formulation it is a statement about complex zeros of Riemann’s zeta function which is the analytical continuation of the series

Much as almost every great problem, Riemann’s hypothesis has many equivalent restatements. Georg Kreisel [25] managed to reformulate it as an assertion about the emptiness of a particular listable set (each element of this set would produce a counterexample to the hypothesis). Respectively, we can construct a Diophantine representation of this set and obtain a particular Diophantine equation

which has no solution if and only if the Riemann hypothesis is true.

It was my share to write Sect. 2 of [12] devoted to reductions of Riemann’s hypothesis and some other famous problems to Diophantine equations, but I failed to present the equation whose unsolvability is equivalent to Riemann’s Hypothesis. Kreisel’s main construction was very general, applicable to any analytical function, and some details of how to transfer it to a Diophantine equation were cumbersome. Luckily, Harold N. Shapiro, a colleague of Martin Davis, came to help and suggested a simpler construction, specific to the zeta function, based on the relationship of Riemann’s hypothesis and distribution of prime numbers, and the corresponding part of Sect. 2 from [12] was written by Martin Davis.

In [37] I present a reductions of Riemann’s hypothesis to Diophantine equations that is a bit simpler that the construction in [12], the simplification was due to certain new explicit constants related to distribution of primes that were obtained at that time in Number Theory.

Thus, Riemann’s hypothesis can be viewed as a very particular case of Hilbert’s tenth problem; such a relationship between it and Hilbert’s eighth problem was not known before the DPRM-theorem was proved.

Hardly one can hope to prove or to disprove Riemann’s hypothesis by examining a corresponding Diophantine equation. On the other hand, such a reduction gives an informal “explanation” of why Hilbert’s tenth problem is undecidable: it would be rather surprising if such a long-standing open problem could be solved by a mechanical procedure required by Hilbert.

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