Approximation of the Riemann Zeta Function via MERA Tensor Networks: A Study of Convergence and Error Bounds Based on Mellin Operators

Abstract

This paper presents a novel computational framework that maps the analytic continuation of the Riemann Zeta function $\zeta(s)$ in the critical strip $0 < \Re(s) < 1$ onto Multiscale Entanglement Renormalization Ansatz (MERA) tensor networks via the Dirichlet Eta function $\eta(s)$ and Mellin integral operators. We establish a formal unitary equivalence between a single-layer MERA coarse-graining step and the discretized Mellin operator (Lemma 3.1). Furthermore, we analytically derive the theoretical infrared fixed point $\gamma_{\text{target}} = \frac{\log(2\pi)}{2\pi^2} \approx 0.093102\dots$ from the critical polygamma curvature of the Riemann functional equation combined with the MERA metric pullback (Theorem 3.2). Finally, we provide exponential error bounds decoupled in network depth and bond dimension (Theorem 3.3), validated against numerical contractions ($\chi=2$) achieving a residual of $1.3 \times 10^{-5}$.