Convergence of the Flint Hills series

What the AI did

The release README says the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result, out of roughly 4,000 problems posed; the output was aggregated into result families and manuscripts and kept if judged significant enough. This family has one manuscript, dated September 24, 2026; the series result is a corollary of its main theorem on the irrationality exponent of pi. The manuscript is credited to 'OpenAI' alone and names no human author. The README's two exceptions to the fixed procedure (the Riemann zeta zero-free region work, whose Re(s) > 11/12 write-up was human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The README also cautions that unformalized results could have issues. OpenAI also released an abridged reasoning summary for this family (reasoning_traces/irrationality-exponent-of-pi.pdf).

Verification

No independent mathematician has checked this yet. Checked here: Corollary 1.2 and Section 5 of the TeX source read against the question; convergence follows from $\mu(\pi)<5/2$ by a short dyadic spacing lemma that the paper proves (Meiburg's criterion is cited too). Everything rests on the unrefereed main theorem $\mu(\pi)=2$. The release's Lean scope note says the Flint Hills consequence is outside the Comparator challenge PiExponent. The solution module OAI.NumberTheory.PiExponent.Main also contains a theorem flint_hills_summable stating summability of $1/(n^3\sin^2 n)$, but it is not a Comparator challenge, so it is not counted here; nothing was rebuilt.

Sources

• CodeOpenAI math release: The irrationality exponent of pi is 2
• Problem recordAlekseyev, On convergence of the Flint Hills series (arXiv 2011)
• OtherAbridged reasoning summary released by OpenAI for this family