A new upper bound for the irrationality exponent of $\zeta (2)$

Abstract

We prove that the irrationality exponent of $\zeta (2)={\pi }^2/6$ satisfies $\mu (\zeta (2))<5.0193784$. The best published bound is 5.095412 (Zudilin, 2014); a bound 5.0495243 with a machine-checked proof was made public in September 2026 by J. Kleid. We use Zudilin's two hypergeometric constructions at a new choice of parameters. The two families of linear forms in 1 and $\zeta (2)$ so obtained coincide for every sufficiently large index: this coincidence, an instance of an identity conjectured by Zudilin, is established by comparing an arithmetic bound for the denominators of the difference with an analytic bound for its size, without creative telescoping. All numerical constants are certified in ball arithmetic.

arXiv comment6 pages. Ancillary files: verify_zeta2.py (certifies all constants in ball arithmetic; requires python-flint and numpy) and verify_zeta2_exact.py (exact rational checks; standard library only)