Modular L-Values Link Apéry Numbers to Sporadic Sequences — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Zagier's interpolation of Apéry numbers for ζ(3) is a critical L-value of a weight-4 modular form; this extends to all six sporadic sequences, linking number-theoretic recurrences to modular symmetries. | MATH: Apéry numbers an = ∑k=0n binomnk2 binomn+kk2 satisfy an = ∑k=0n binomnk2 binomn+kk2 (recurrence: (n+1)3 an+1 − (2n+1)(17n2+17n+5)an + n3 an−1 = 0). Zagier's interpolation: A(t) = ∑n≥0 an tn (or a shifted/generating variant) evaluated at special t yields L(f,2) for a weight-4 modular form f (e.g., associated to elliptic curve of conductor 37, L(f,2) = 4π2/37 · (rational)). The six sporadic sequences (Zagier's list: A,B,C,D,E,F with parameters (a,b,c) in the recurrence n3 un = (a n2 + b n + c) un−1 − n3 un−2) all have interpolations expressible as L(f,2) for weight-4 modular forms with rational coefficients. | CONNECTION: Weight-4 mod
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- Modular L-Values Link Zagier's Sporadic Sequences to Apéry Numbers — E8 Intelligence Research
- Φ-Defect Cascades Encode Markov Numbers on Modular Cusps — E8 Intelligence Research
- Mellin Transform Bridges Modular Forms and Apéry Numbers via L-Functions — E8 Intelligence Research
- The 12-Periodicity Bridge: Silver Ratio, Sexagesimal, and 8-Fold Modular E8 Structure — E8 Intelligence Research
- Mediant Trees, Farey Sequences, and Rational Enumeration via Continued Fractions — E8 Intelligence Research
BibTeX
@misc{caldin2026modular,
title = {{Modular L-Values Link Apéry Numbers to Sporadic Sequences — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23431162},
url = {https://zenodo.org/records/23431162},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Modular L-Values Link Apéry Numbers to Sporadic Sequences — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23431162
UR - https://zenodo.org/records/23431162
AB - FINDING: Zagier's interpolation of Apéry numbers for ζ(3) is a critical L-value of a weight-4 modular form; this extends to all six sporadic sequences, linking number-theoretic recurrences to modular symmetries. | MATH: Apéry numbers a_n = ∑_k=0^n binomnk^2 binomn+kk^2 satisfy a_n = ∑_k=0^n binomnk^2 binomn+kk^2 (recurrence: (n+1)^3 a_n+1 − (2n+1)(17n^2+17n+5)a_n + n^3 a_n−1 = 0). Zagier's interpolation: A(t) = ∑_n≥0 a_n t^n (or a shifted/generating variant) evaluated at special t yields L(f,2) for a weight-4 modular form f (e.g., associated to elliptic curve of conductor 37, L(f,2) = 4π^2/37 · (rational)). The six sporadic sequences (Zagier's list: A,B,C,D,E,F with parameters (a,b,c) in the recurrence n^3 u_n = (a n^2 + b n + c) u_n−1 − n^3 u_n−2) all have interpolations expressible as L(f,2) for weight-4 modular forms with rational coefficients. | CONNECTION: Weight-4 mod Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -