Mellin Transform Bridges Modular Forms and Apéry Numbers via L-Functions — E8 Intelligence Research

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 \( \mathcal{M}[f](s) = \int_0^\infty f(x) x^{s-1} dx \); for modular form \( f(\tau) = \sum a_n q^n \), \( L(f,s) = \mathcal{M}[f(iy)](s) \) up to gamma factors; Apéry numbers \( A_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2 \) satisfy \( \sum A_n q^n = \) modular form of weight 4 (for \( \zeta(3) \) case); Riemann zeta \( \zeta(s) = \frac{1}{\Gamma(s)} \int_0^\infty \frac{x^{s-1}}{e^x-1} dx \). | CONNECTION: The critical line \( \Re(s)=1/2 \) for zeta — the value 0.5 is not directly in the golden ratio family, but the functional equation \( s \leftrightarrow 1-s \) has fixed point \( s=1/2 \), mirroring the self-dual symmetry of \( \phi \) (since \( \phi-1 = 1/\phi \)). The modular group \( SL(2,\mathbb{Z}) \) has cusp at \( \infty \) 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