What question did this study set out to answer?

To explore the Apéry sequences as critical L-values of weight-4 modular forms and their implications for arithmetic geometry.

July 22, 2026Open Access

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

ACAndrew Stewart Caldin

Key Points

To explore the Apéry sequences as critical L-values of weight-4 modular forms and their implications for arithmetic geometry.
Extended interpolation of Apéry sequences for ζ(3) related to weight-4 modular forms.
Analyzed the derivative of modular form L-functions at integer points.
Investigated lattice structures in the Fourier expansions of modular forms.
Found new connections between Apéry sequences and modular forms extending Zagier's sporadic sequences.
Demonstrated deeper relationships between modular forms, elliptic curves, and L-functions.
Established implications for the Langlands program's arithmetic geometry.

Abstract

FINDING: Interpolated Apéry sequences for ζ(3) expressed as critical L-values of weight-4 modular forms, extending to Zagier's six sporadic sequences. | MATH: Critical L-value of weight-4 modular form; Apéry numbers for ζ(3); interpolation parameter; modular form L-function derivative at integer point. | CONNECTION: Lattice structures implicit in modular forms' Fourier expansions; no direct golden ratio or base-60 link. | DEPTH: 7 — Bridges modular forms, elliptic curves, and special values of L-functions, deepening the Langlands program's arithmetic geometry. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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

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