Interpolated Apéry-like sequences extend Zagier's link between ζ(3) and modular L-values. — E8 Intelligence Research

Abstract

FINDING: Interpolated Apéry-like sequences express critical L-values of modular forms, extending Zagier's connection between ζ(3) and weight-4 modular forms. MATH: - Apéry numbers for ζ(3): \( a_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2 \) - Zagier's interpolation: \( A(t) = \sum_{n=0}^\infty a_n t^n \) related to critical L-value \( L(f,2) \) for weight-4 modular form \( f \). - Generalization: Interpolated sequences for Zagier's six sporadic sequences yield \( L(f, k) \) for modular forms of varying weights. - Hypergeometric representation: \( {}_4F_3 \) or \( {}_3F_2 \) series with rational parameters, often with argument 1 or -1. CONNECTION: - No direct geometric ratios (0.382, 0.618, 1.618) appear. - However, the underlying modular forms are associated with elliptic curves and lattices, which have crystallographic symmetry (e.g., root lattice \( E_8 \) in weight-4 case for ζ(3) via modularity). - The critical L-values are evaluated at integer points, linking Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com