A Nonlinear Deficiency Identity for the Riemann Zeta Function with Optimal Approximation Rates

TL;DR

This paper presents a novel nonlinear framework for approximating the Riemann zeta function, providing structural identities, bias correction, and optimal convergence rates validated by numerical experiments.

Contribution

It introduces a deficiency-based representation and approximation method that offers a unified nonlinear perspective on zeta function approximation, extending classical techniques.

Findings

Derived an exact identity relating zeta values via deficiency functional.

Proved convergence rates for corrected estimators with bias removal.

Numerical experiments confirm theoretical convergence and extend to spectral zeta functions.

Abstract

We introduce a deficiency-based representation and approximation framework for values of the Riemann zeta function. The method is based on comparing two nonlinear accumulation mechanisms: global transformation of a base partial sum and local transformation of each term. Their gap defines a cumulative deficiency functional that yields the exact identity \[ \zeta(q)=\zeta(p)^{q/p}-D_{\infty}^{(p,q)}, \qquad q>p>1. \] This converts zeta approximation into estimation of a nonlinear deficit. We derive corrected estimators that remove first-order bias and prove the convergence law \[ B_n^{(p,q)}-\zeta(q)=O\!\left(n^{-\min(2p-2,q-1)}\right). \] For odd targets, suitable choices of the base exponent recover the natural truncation rate while preserving the structural identity. Numerical experiments for $\zeta(3),\zeta(5),\zeta(7)$ confirm theory, demonstrate strong finite-sample behavior, and illustrate extension to spectral zeta functions. The contribution is structural rather than replacing classical Euler--Maclaurin methods: we provide a unified nonlinear viewpoint on zeta approximation, convexity-induced correction terms, and tunable approximation families.