Modular Forms Realize Apéry-Type Sequences as Critical L-Values — E8 Intelligence Research

FINDING: Interpolated Apéry-type sequences for ζ(3) and Zagier's six sporadic sequences are realized as critical L-values of weight-4 modular forms, bridging hypergeometric series and modular form arithmetic. | MATH: Apéry numbers \(A_n = ∑ₖ₌₀^n {n}{k}^2 {n+k}{k}^2\) satisfy \(A_n ~ {(1+√2)⁴ⁿ⁺²}{(2π)3/2 n3/2}\); Zagier's interpolation yields \(A_n(t)\) such that \(A_n(t) = L_f(2, t)\) for a weight-4 modular form \(f\), with critical value at \(s=2\). Sporadic sequences: \(s(n) = ∑ₖ₌₀^n {n}{k}^3\) (Franel), \(s(n) = ∑ₖ₌₀^n {n}{k}^2 {2k}{k}\) (Domb), etc. — all six map to \(L\)-values of modular forms of weight 4. | CONNECTION: The critical \(L\)-value \(L_f(2)\) relates to periods of elliptic curves (weight-2) lifted to weight-4 via Rankin–Selberg; the ratio of successive Apéry numbers tends to \((1+√2)^4 ≈ 33.97\), whose square root is \(1+√2 ≈ 2.414\), linked to the silver ratio — a metall Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com