Hello AIT mailing list,
A recent exchange with a Swedish number theorist, on Cramér's conjecture, motivated
me to simplify the presentation of the Monte Carlo Hypothesis for probabilistic number
theorists and clarify its implications for both Cramér's conjecture and the Riemann Hypothesis.
I'd like to add a few points that are important but have not been explicitly stated.
The main technical innovation is to reformulate certain propositions in analytic number theory
that many take for granted, such as the Prime Number Theorem, in order to carefully assess
their information-theoretic implications. Philosophically, I am partly motivated by Gregory Chaitin's
complexity-bound on Gödel numbers whose truth-value is decidable. This suggests that experimental
mathematics is eventually necessary for sufficiently-complex mathematical propositions.
Another key motivation comes from the physics of the Riemann Hypothesis but this is not
something that can be easily summarised in a few pages.
p.s. a colleague that does machine learning at Oxford encouraged me to submit this as a neurips challenge.
If that goes through, in 2022, I will write a book on experimental mathematics for AI researchers with a focus
on both open problems as well as algorithmic methods. You may be interested in the work of Yang-Hui He: