A Nonlinear Deficiency Identity for the Riemann Zeta Function with Optimal Approximation Rates
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914525768318976 |
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| author | Mohammady, Meisam |
| author_facet | Mohammady, Meisam |
| contents | 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 \[ ζ(q)=ζ(p)^{q/p}-D_{\infty}^{(p,q)}, \qquad q>p>1. \] This converts zeta approximation into estimation of a nonlinear deficit. We derive corrected estimators that remove first-order bias and prove the convergence law \[ B_n^{(p,q)}-ζ(q)=O\!\left(n^{-\min(2p-2,q-1)}\right). \] For odd targets, suitable choices of the base exponent recover the natural truncation rate while preserving the structural identity. Numerical experiments for $ζ(3),ζ(5),ζ(7)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16530 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Nonlinear Deficiency Identity for the Riemann Zeta Function with Optimal Approximation Rates Mohammady, Meisam General Mathematics 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 \[ ζ(q)=ζ(p)^{q/p}-D_{\infty}^{(p,q)}, \qquad q>p>1. \] This converts zeta approximation into estimation of a nonlinear deficit. We derive corrected estimators that remove first-order bias and prove the convergence law \[ B_n^{(p,q)}-ζ(q)=O\!\left(n^{-\min(2p-2,q-1)}\right). \] For odd targets, suitable choices of the base exponent recover the natural truncation rate while preserving the structural identity. Numerical experiments for $ζ(3),ζ(5),ζ(7)$ 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. |
| title | A Nonlinear Deficiency Identity for the Riemann Zeta Function with Optimal Approximation Rates |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2604.16530 |