Modular L-Values Link Zagier's Sporadic Sequences to Apéry Numbers — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The search results are a mix of generic recurrence-solving pedagogy and one substantive research paper linking interpolated Apéry numbers to critical L-values of modular forms. The core mathematical insight is that Zagier's interpolation of Apéry numbers (for ζ(3)) is not an isolated combinatorial curiosity but a critical L-value of a weight-4 modular form, and this extends to all six of Zagier's sporadic sequences.
MATH: - Apéry numbers: an = ∑k=0n binomnk2 binomn+kk2, satisfying recurrence (n+1)3 an+1 = (2n+1)(17n2+17n+5) an − n3 an−1. - Characteristic polynomial of the recurrence (in the limit of large n, after scaling): roots are λ = 17 + 12√2 ≈ 33.97 and its reciprocal 17 − 12√2 ≈ 0.0294. Note 17 + 12√2 = (1+√2)4, and (1+√2) is the silver ratio. - Modular form connection: The interpolated Apéry numbers At(n) (Zagier's t-deformation) are shown to
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026modular,
title = {{Modular L-Values Link Zagier's Sporadic Sequences to Apéry Numbers — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635248},
url = {https://zenodo.org/records/23635248},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Modular L-Values Link Zagier's Sporadic Sequences to Apéry Numbers — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635248
UR - https://zenodo.org/records/23635248
AB - FINDING: The search results are a mix of generic recurrence-solving pedagogy and one substantive research paper linking interpolated Apéry numbers to critical L-values of modular forms. The core mathematical insight is that Zagier's interpolation of Apéry numbers (for ζ(3)) is not an isolated combinatorial curiosity but a critical L-value of a weight-4 modular form, and this extends to all six of Zagier's sporadic sequences. MATH: - Apéry numbers: a_n = ∑_k=0^n binomnk^2 binomn+kk^2, satisfying recurrence (n+1)^3 a_n+1 = (2n+1)(17n^2+17n+5) a_n − n^3 a_n−1. - Characteristic polynomial of the recurrence (in the limit of large n, after scaling): roots are λ = 17 + 12√2 ≈ 33.97 and its reciprocal 17 − 12√2 ≈ 0.0294. Note 17 + 12√2 = (1+√2)^4, and (1+√2) is the silver ratio. - Modular form connection: The interpolated Apéry numbers A_t(n) (Zagier's t-deformation) are shown to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -