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

Meisam Mohammady

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Pith reviewed 2026-05-10 08:26 UTC · model grok-4.3

classification 🧮 math.GM

keywords zetaapproximationnonlinearidentitybasedeficiencyfunctionriemann

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The pith

Derives the identity ζ(q) = ζ(p)^{q/p} - D_∞^{(p,q)} for q > p > 1 and proves corrected estimators converge at rate O(n^{-min(2p-2, q-1)}).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The zeta function adds up terms like 1 over n to the power s for different s values. The authors compare two ways of building these sums: one that transforms the entire partial sum at once, and another that transforms each individual term. The difference between these two builds up into a deficiency quantity. Setting the two builds equal except for this deficiency produces an exact relation between zeta at one exponent and zeta at another. They then show how to estimate the deficiency from finite sums and correct for bias, yielding error rates that depend on the chosen exponents and improve as more terms are included. For certain odd targets the rate matches the best possible from simple truncation.

Core claim

The exact identity ζ(q)=ζ(p)^{q/p}-D_∞^{(p,q)} for q>p>1 together with the convergence law B_n^{(p,q)}-ζ(q)=O(n^{-min(2p-2,q-1)}).

Load-bearing premise

That the gap between the global transformation of the base partial sum and the local transformation of each term defines a well-behaved cumulative deficiency functional whose estimation yields the stated identity and rates without additional hidden assumptions on the analytic continuation or convergence properties.

Figures

Figures reproduced from arXiv: 2604.16530 by Meisam Mohammady.

**Figure 1.** Figure 1: Absolute approximation error for ζ(3). The corrected deficiency estimator with p = 2 follows the predicted O(n −2 ) rate and improves finite-sample accuracy relative to the uncorrected form. 4.2 Experiment I: Approximation of ζ(3) We begin with the first odd target: q = 3, p = 2. This case is canonical because p = 2 is both explicit and optimal: p∗ = q + 1 2 = 2 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗

**Figure 2.** Figure 2: Absolute approximation error for ζ(5). Choosing p = 4 substantially improves convergence relative to the universal base p = 2. 4.4 Experiment III: Asymptotic Verification for ζ(5) To verify the asymptotic model beyond slope estimates, [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Figure 3: Scaled error n 4 |B (4,5) n − ζ(5)|. Stabilization to a plateau confirms the asymptotic law Cn−4 . The emergence of a stable plateau verifies the full asymptotic relation B (6,7) n − ζ(7) ∼ Cn−6 . Minor deviations at very large n are attributable to double-precision limitations rather than methodological failure. 4.7 Experiment VI: Spectral Zeta Functions Finally, we test generalized spectra λk = k α , α ∈… view at source ↗

**Figure 4.** Figure 4: Absolute approximation error for ζ(7). The optimized estimator with p = 6 achieves the predicted sixth-order decay [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

**Figure 5.** Figure 5: Scaled error n 6 |B (6,7) n − ζ(7)|. The plateau confirms sixth-order convergence before floating-point saturation. [7] Jonathan M. Borwein, David H. Bailey, and Roland Girgensohn. Experimentation in Mathematics: Computational Paths to Discovery. A K Peters, 2004. [8] Konrad Knopp. Theory and Application of Infinite Series. Dover, 1990. [9] Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charl… view at source ↗

**Figure 6.** Figure 6: Spectral experiments for λk = k α, α ∈ {2, 3, 4}. The deficiency framework remains stable across multiple spectral growth regimes. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean exact identity for zeta(q) via a deficiency gap plus a specific convergence rate for bias-corrected partial sums, but the advance is mostly a reformulation with moderate practical payoff.

read the letter

The one or two things your colleague should know are the exact identity ζ(q) = ζ(p)^{q/p} - D_∞^{(p,q)} for q > p > 1, which comes from comparing global and local nonlinear accumulations, and the proved rate B_n^{(p,q)} - ζ(q) = O(n^{-min(2p-2, q-1)}) for the corrected estimators. Those are the concrete claims that stand out from the abstract. The paper does a solid job spelling out how this turns approximation into deficit estimation, deriving the first-order bias corrections, and showing that suitable p recovers the natural truncation rate for odd targets. The numerical checks on ζ(3), ζ(5), and ζ(7) line up with the theory and the extension note to spectral zeta functions is a reasonable broadening. It is honest in calling the contribution structural rather than a replacement for Euler-Maclaurin. On the soft spots, the identity looks like it follows almost by definition once the deficiency functional is introduced, so the novelty is narrower than the framing suggests. The convergence proof is asserted but the abstract gives no detail on how the analytic continuation or the accumulation is controlled, leaving room for hidden assumptions. The free parameter p requires choice and the rate is not obviously superior to existing methods for general use. The numerics are confirmatory rather than comparative stress tests. This is for analytic number theorists who enjoy nonlinear viewpoints on zeta approximation and want tunable families. It has enough explicit statements and falsifiable numerics to deserve a serious referee who can check the derivations, even if the overall impact stays modest. I would send it to peer review.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the standard definition of zeta as an infinite sum for Re(s)>1 and on the postulate that the difference between global and local nonlinear transformations yields a usable deficiency functional; no free parameters are fitted to data and no new entities beyond the defined deficiency are introduced.

free parameters (1)

base exponent p
Chosen freely with p>1 to define the family of identities and the resulting convergence rate; not fitted to data.

axioms (1)

standard math Zeta function defined by the Dirichlet series sum n^{-s} for Re(s)>1
Invoked to justify the partial sums and the global/local transformations.

invented entities (1)

deficiency functional D_∞^{(p,q)} no independent evidence
purpose: Captures the exact gap between global and local nonlinear accumulation mechanisms
Defined directly as the difference; no independent falsifiable prediction outside the identity itself.

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Works this paper leans on

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