Mellin Transform Bridges Modular Forms and Apéry Numbers via L-Functions — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Mellin transform bridges Dirichlet series and modular forms via integral representations, with Apéry numbers emerging as special values of modular forms' L-functions. | MATH: Mellin transform ℳ[f](s) = ∫0∞ f(x) xs−1 dx; for modular form f(τ) = ∑ an qn, L(f,s) = ℳ[f(iy)](s) up to gamma factors; Apéry numbers An = ∑k=0n binomnk2 binomn+kk2 satisfy ∑ An qn = modular form of weight 4 (for ζ(3) case); Riemann zeta ζ(s) = 1/Γ(s) ∫0∞ (xs−1)/(ex−1) dx. | CONNECTION: The critical line ℜ(s)=1/2 for zeta — the value 0.5 is not directly in the golden ratio family, but the functional equation s ↔ 1−s has fixed point s=1/2, mirroring the self-dual symmetry of ϕ (since ϕ−1 = 1/ϕ). The modular group SL(2,ℤ) has cusp at ∞ and fundamental domain with angles \( \pi/3, \pi
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026mellin,
title = {{Mellin Transform Bridges Modular Forms and Apéry Numbers via L-Functions — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23409887},
url = {https://zenodo.org/records/23409887},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mellin Transform Bridges Modular Forms and Apéry Numbers via L-Functions — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23409887
UR - https://zenodo.org/records/23409887
AB - FINDING: Mellin transform bridges Dirichlet series and modular forms via integral representations, with Apéry numbers emerging as special values of modular forms' L-functions. | MATH: Mellin transform ℳ[f](s) = ∫_0^∞ f(x) x^s−1 dx; for modular form f(τ) = ∑ a_n q^n, L(f,s) = ℳ[f(iy)](s) up to gamma factors; Apéry numbers A_n = ∑_k=0^n binomnk^2 binomn+kk^2 satisfy ∑ A_n q^n = modular form of weight 4 (for ζ(3) case); Riemann zeta ζ(s) = 1/Γ(s) ∫_0^∞ (x^s−1)/(e^x−1) dx. | CONNECTION: The critical line ℜ(s)=1/2 for zeta — the value 0.5 is not directly in the golden ratio family, but the functional equation s ↔ 1−s has fixed point s=1/2, mirroring the self-dual symmetry of ϕ (since ϕ−1 = 1/ϕ). The modular group SL(2,ℤ) has cusp at ∞ and fundamental domain with angles \( \pi/3, \pi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -