What question did this study set out to answer?

Examine the connection between interpolated Apéry-like sequences and critical L-values of modular forms.

July 24, 2026Open Access

Interpolated Apéry-like Sequences Connect Six Sporadic Cases to Modular L-Values — E8 Intelligence Research

Key Points

Examine the connection between interpolated Apéry-like sequences and critical L-values of modular forms.
Analyzed Apéry numbers for ζ(3) and derived recurrence relations.
Generalized findings to six sporadic sequences associated with modular forms of weights 2, 3, and 4.
Explored the relationship between modular forms and elliptic curves.
Identified critical L-values corresponding to the six sporadic sequences linked to root systems (A, D, E).
Showed that Apéry-like sequences extend Zagier’s previous work with ζ(3), linking deeper mathematical concepts.
Demonstrated that critical L-values relate to the arithmetic structure outlined in the Birch–Swinnerton-Dyer conjecture.

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 to six sporadic sequences. | MATH: Apéry numbers for ζ (3) satisfy recurrence (n+1) ³ u₍+₁ = (2n+1) (17n²+17n+5) uₙ - n³ u₍-₁. Interpolated version yields critical L-value of weight-4 modular form. Generalization: six sporadic sequences (Zagier's list) interpolate to critical L-values of modular forms of weights 2, 3, 4. Key constants: ζ (3) ≈ 1. 2020569, L-values at s = 1, 2, 3. | CONNECTION: No direct geometric ratio (0. 382, 0. 618, 1. 618) or base-60 appears. However, modular forms are deeply tied to elliptic curves and lattice symmetries (SL (2, Z) action on upper half-plane). Critical L-values encode arithmetic of elliptic curves (Birch–Swinnerton-Dyer conjecture). The six sporadic sequences correspond to root systems of type A, D, E (ADE classification) — crystallographic symmetry groups. | DEPTH: 8 — Bridges number theory (Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com

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