Zagier's Interpolation of Apéry Numbers via Modular L-Values for Sporadic Sequences — E8 Intelligence Research

FINDING: Zagier's interpolation of Apéry numbers for ζ(3) as critical L-values of weight-4 modular forms, extended to all six sporadic sequences. MATH: - Apéry numbers: \( A_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2 \), satisfying \( (n+1)^3 A_{n+1} - (34n^3 + 51n^2 + 27n + 5) A_n + n^3 A_{n-1} = 0 \). - Zagier's interpolation: \( A_n(t) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k} \binom{n+k}{k} t^k \) (or similar deformation) evaluated at \( t = -1 \) gives \( A_n(-1) \) linked to \( L(f, 2) \) for a weight-4 newform \( f \). - Critical L-value: \( L(f, 2) = \frac{(2\pi)^2}{N^{3/2}} \cdot \text{algebraic number} \) (up to rational factors), with \( N \) the conductor. - The six sporadic sequences (Zagier's \( A, B, C, D, E, F \)) all satisfy second-order recurrences with polynomial coefficients; their interpolations yield \( L \)-values of weight-4 forms with CM or non-CM. - Key constants: The recurrence coefficients involve integers like 34, 51, 27, 5 (for ζ(3) ca Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com