Updates to arXiv:2610.27799 that will appear in the next arXiv version.

Linear independence of 1, ζ(2) and L(2,χ₋₃)

We prove that 1, ζ(2) and L(2,χ₋₃) are linearly independent over ℚ. This is the main result of Calegari, Dimitrov and Tang, The linear independence of 1, ζ(2), and L(2,χ₋₃) (arXiv:2408.15403), who proved it using arithmetic holonomy bounds.

Our proof is Apéry-style: a single explicit recurrence of order three, coming from a CMF trajectory, gives the rational approximations that prove the result.

The notebook proves that the dual Meijer G state is recessive, identifies both limits through the Dual Convergence Theorem of the paper, derives the exponential error rate, and certifies the arithmetic bounds with exact and interval arithmetic. For every fixed nonzero integer combination, the approximation exponent satisfies δ > 0.0001.

The notebook, the written arguments and the certificates are in the paper’s supplementary code: new-directions-in-irrationality/additional/l2chi3_zeta2.

Last updated: October 2026.