Interpolated Apéry Numbers as Critical L-Values of Weight-4 Modular Forms — E8 Intelligence Research

FINDING: Interpolated Apéry numbers for ζ (3) are critical L-values of weight-4 modular forms, extending Zagier's result to six sporadic sequences. MATH: - Apéry numbers \ (Aₙ = ₊=₀ⁿ nk² n+kk² \) satisfy \ (Aₙ (3) \) rational approximations. - Zagier's interpolation: \ (Aₙ (t) = ₊=₀ⁿ nk² n+kk² tᵏ \). For \ (t=1 \), recovers Apéry numbers. For \ (t=-1 \), yields critical L-value of a weight-4 modular form. - Generalization: Each of Zagier's six sporadic sequences (e. g. , \ (s₆, s₇, s₁₀, s₁₈ \) ) interpolates to a modular form L-value at \ (s=1 \) (or \ (s=0 \) ). - E8 theta series: Theta function \ (₄₈ (q) = 1 + 240₍₁ ₃ (n) qⁿ \) is a modular form of weight 4 for \ (SL₂ (Z) \). Its Fourier coefficients involve divisor sums \ (₃ (n) \). CONNECTION: - E8 root system: 240 roots, Coxeter number 30, Weyl group order 696729600. Theta series coefficients \ (Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com