The irrationality exponent of pi is 2
Source: github.com/openai/math/tree/main/preprints/The-irrationalit…
We prove the conjecture that the irrationality exponent of π is 2. As a consequence, the classical Flint–Hills series $\sum_{n\ge1}1/(n^3\sin^2 n)$ converges, with angles in radians.
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Apparently, the so-called internal model of OpenAI has recently established that the irrationality exponent of $\pi$ is equal to $2$ . It was well-known that the convergence of the Flint Hill series would follow from an irrationality exponent of $\pi$ less than $5/2$ (see, for instance, the…
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A Brief Digest of Some New OpenAI Results | Persiflagegaloisrepresentations.orgFrank Calegari
the close similarity between the arguments here and Roth’s argument for algebraic irrationals. Again, this is something that I hope we shall address at our upcoming conference. Broadly speaking, for the problems in the Langlands program that I am interested in, while OpenAI did an amazing job,…