A Gaussian-Perron Prime-Side Defect and Local Profiles Near Critical-Line Zeros of the Riemann Zeta Function

Netzer Moriya

A Gaussian-Perron Prime-Side Defect and Local Profiles Near Critical-Line Zeros of the Riemann Zeta Function

T0 review · 0 major / 5 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read A Gaussian-smoothed prime force localizes near each critical-line zero of zeta to a universal logarithmic profile, under RH and a mild width condition.

desk verdict Clean RH-conditional local profile for a new Gaussian-smoothed prime-force defect; standard contour work, useful diagnostic language, no RH breakthrough. read the letter →

arxiv 2607.04316 v2 pith:7JHX6BBR submitted 2026-07-05 math.NT

Netzer Moriya This is my paper · ORCID

classification math.NT MSC 11M2611M0611N05

keywords RiemannzetafunctionexplicitformulaGaussian–Perronsmoothingprime-forcedefectlocalzeroprofilesanisotropicdampingcriticalline

topics The Riemann Hypothesis The Twin Prime Conjecture

open problems The Riemann Hypothesis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper builds a Gaussian–Perron smoothed prime-side field and compares it with zeta-prime over zeta, weighted by horizontal distance from the critical line. That defect converts the classical explicit formula into a local probe of zero geometry. On the logarithmic scale s equals rho-zero plus lambda over log X, a simple critical-line zero contributes an exact universal profile minus a times the real part of e to the minus lambda over lambda. An anisotropic damping law, controlled by a quadratic functional Q-alpha, separates the selected zero from the surrounding zero cloud and from the pole. Under the Riemann Hypothesis and an explicit lower bound on the free smoothing width alpha, every non-selected contribution is exponentially small, so the full defect equals that selected-zero profile up to an O of 1 over log X plus an exponentially small remainder. A direct prime-power numerical check near the first zero reproduces the finite-X profile to high accuracy. The result gives a concrete local diagnostic: the smoothed primes “see” each simple critical-line zero through a fixed logarithmic shape once the width is large enough to damp the pole.

What carries the argument

The Gaussian–Perron prime-force defect Δ_{X,α}(s)=(\sigma−1/2) Re(P_{X,α}(s)−ζ′/ζ(s)), whose kernel supplies both an error-function prime weight and an anisotropic zero-side damping functional Q_α that localizes the explicit formula to a single selected residue.

What would settle it

Compute the truncated prime-side defect near a known simple critical-line zero for several X and α that satisfy the pole-damping bound, and check whether the pointwise discrepancy from −a Re(e^{−λ}/λ) decays like 1/log X plus an exponentially small term; a persistent larger residual would refute the localization.

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Extended reading notes

Core claim

Assuming the Riemann Hypothesis and a mild lower bound on the Gaussian smoothing width relative to a fixed simple critical-line zero, the full Gaussian–Perron prime-force defect near that zero equals the universal selected-zero profile −a Re(e^{−λ}/λ) plus an error that is O(1/log X) plus exponentially small, uniformly on compact sets of the logarithmic displacement λ away from zero.

Load-bearing premise

The free smoothing width must be large enough relative to the fixed zero’s height so that the pole contribution is exponentially damped, and the shifted vertical contour must obey a standard logarithmic-derivative bound.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The punchline is simple: under RH and an explicit lower bound on the Gaussian width α relative to a fixed simple critical-line zero, the full prime-force defect Δ_{X,α} equals the exact selected-zero residue profile plus an exponentially small remainder. That is Theorem 7.6. Everything else is scaffolding for that statement.

What is actually new is the object itself. The Gaussian–Perron kernel produces both an error-function prime weight and the anisotropic damping functional Q_α. The selected-zero identities (Lemma 5.2, Theorems 5.3–5.4) are elementary residue calculations, but they give a clean universal profile on the log-X scale and a linear spike for a hypothetical off-line zero. The finite-window damping certificate and the geometry of the Q_α = 0 boundary are the paper’s real contribution to the explicit-formula toolkit. The appendices write the large-height Gaussian tails and the shifted-contour majorant carefully; under RH the non-selected zeros sit on the critical line so the damping is automatic and the estimates are standard Riemann–von Mangoldt plus log-derivative bounds.

Soft spots are real but limited. Full-defect control off the critical line is deliberately left open (Remark 5.5). The pole-damping condition forces α large enough relative to γ_{0}; that is a free-parameter hypothesis, not a hidden gap. The numerical check near the first zero is only illustrative and no code is shipped. None of this breaks the theorems as stated.

This is for people who already work with explicit formulas, hybrid Euler–Hadamard products, or horizontal-force reformulations of RH. It does not resolve RH and does not give a global zero law. It does give a local diagnostic language and a numerically accessible prime-side probe. I would send it to referees; the math is grounded enough to deserve the time. Worth reading if you care about local zero geometry; skip if you only want unconditional global statements.

Referee Report

0 major / 5 minor

Summary. The paper constructs a Gaussian–Perron smoothed prime field P_{X,α} and the associated horizontal prime-force defect Δ_{X,α}(s) = (σ−1/2) Re(P_{X,α}(s) − ζ′/ζ(s)). A contour shift yields an explicit formula (Theorem 3.3) whose zero-side terms are controlled by the anisotropic damping functional Q_α. On the logarithmic scale s = ρ₀ + λ/log X the selected residue is evaluated exactly (Lemma 5.2), producing a universal bounded profile at a simple critical-line zero (Theorem 5.3) and a linear-in-log-X spike for a hypothetical off-line zero (Theorem 5.4). Under explicit finite-window damping, pole-damping, and shifted-contour hypotheses the nonlocal remainder is shown to be exponentially small (Theorem 7.4); under RH and the pole-damping lower bound on α the same conclusion holds for every fixed simple critical-line zero (Theorem 7.6). A direct prime-side numerical check near the first zero is supplied for illustration.

Significance. If the estimates hold, the work supplies a clean local diagnostic that isolates the contribution of a single critical-line zero to a Gaussian-smoothed prime-side force, with an explicit anisotropic damping boundary for the remaining zero cloud. The selected-zero identities are elementary residue calculations, the RH-conditional localization rests on standard zero-counting and log-derivative bounds once pole damping is imposed, and the numerical check is reproducible from the truncated erfc-weighted prime-power sum. The framework is therefore a useful addition to the local theory of explicit formulae and horizontal-force interpretations of ξ′/ξ, even though it does not resolve RH itself and leaves off-line full-defect control open.

minor comments (5)

In Definition 2.1 and Proposition 2.2 the interchange of sum and integral is justified by absolute convergence on Re z = c, but a one-line reference to the standard majorant for |ζ′/ζ| on Re s > 1 would make the argument self-contained for non-specialists.
Remark 5.5 correctly flags that Theorem 5.4 controls only the isolated residue; a short cross-reference in the introduction would prevent readers from over-interpreting the off-line linear spike as a full-defect statement.
Figure 1 and Table 1 report excellent numerical agreement, yet the truncation N is chosen ad hoc; a brief remark on how N scales with X and α would strengthen the reproducibility claim.
The notation P_{X,α} is used both for the smoothed prime field and for the pole term in the explicit formula (Eqs. (16) and (39)); renaming the pole contribution would remove a minor ambiguity.
Appendix A3 visualizes only the residue surface; a sentence clarifying that the full defect includes the nonlocal remainder E would avoid any visual over-reading of the off-line panel.

Circularity Check

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No circularity: selected-zero profile is an elementary residue evaluation; full-defect localization follows from the explicit formula under stated external hypotheses (RH + pole-damping + contour bounds).

full rationale

The derivation chain is self-contained and non-circular. The prime-force defect is defined by a Gaussian–Perron contour integral (Def. 2.1, Def. 2.3). Contour shift yields the exact zero-side formula (Thm. 3.3) by the residue theorem plus standard vertical/horizontal estimates (Lemmas 3.1–3.2). The selected-zero contribution is then evaluated by direct substitution s = ρ₀ + λ/Y (Lemma 5.2), producing the universal critical-line profile of Thm. 5.3 by elementary algebra; the leading term is independent of the free smoothing parameter α. Nonlocal remainder control (Thms. 7.4 and 7.6) invokes only the classical Riemann–von Mangoldt count, Gaussian tail estimates, the explicit pole-damping inequality on α relative to a fixed ordinate, and a logarithmic-derivative bound on a shifted vertical line—all stated as hypotheses or supplied by RH. No parameter is fitted to data and then re-labeled a prediction; α and X remain free method parameters subject to an explicit inequality. There are no load-bearing self-citations, no uniqueness theorems imported from the author, and no renaming of a known empirical pattern. The numerical check in §8 is pure verification of the already-derived finite-X residue formula against a truncated prime sum, not a circular fit. The RH-conditional claim therefore stands or falls with its external assumptions, not with any internal definitional loop.

Assumptions & free parameters 3 free parameters · 6 assumptions · 2 invented entities

Central RH-conditional claim rests on standard complex analysis and zeta estimates, the external RH assumption, simplicity of the fixed zero, and method parameters (α,X,c,d) with an explicit pole-damping inequality. No data-fitted constants. Invented objects are definitional analytic constructions, not physical postulates.

free parameters (3)

Gaussian width α
Free smoothing parameter of the kernel H_{X,α}. For Theorem 7.6 it must exceed 1/√(2(γ₀²−1/4)) to damp the pole; not fitted to data, but chosen by the user of the method.
Smoothing scale X (Y=log X)
Cutoff center of the error-function prime weight and the logarithmic local scale s=ρ₀+λ/Y. Asymptotic parameter sent to infinity; not fitted.
Contour shifts c,d
Explicit-formula contour parameters with constraints c>1/2 and 0<d<min{1/2,1/α²} in the RH theorem; technical choices that enable residue capture and Gaussian decay on the left line.

assumptions (6)

standard math Classical residue theorem / contour shift for ζ′/ζ against the Gaussian–Perron kernel (Theorem 3.3).
Standard explicit-formula mechanism; used throughout §§3–7.
standard math Riemann–von Mangoldt zero counting and logarithmic-derivative bounds away from zeros/pole (Lemmas 3.1–3.2, A1.1).
Standard zeta estimates cited via Titchmarsh/Montgomery–Vaughan.
domain assumption Riemann Hypothesis: all non-trivial zeros have Re ρ=1/2 (Theorem 7.6).
External unproved hypothesis that makes finite-window damping automatic and justifies the shifted-line bound in the critical strip.
domain assumption Selected zero ρ₀ is simple and fixed with γ₀>0.
Used for the residue formula and nearest-neighbor gap in Appendix A2.
ad hoc to paper Pole-damping inequality 1/2+α²(1/4−γ₀²)<0 for the fixed zero.
Method-specific sufficient condition (Eq. 90) ensuring exponential suppression of the s=1 residue in the local window.
domain assumption Shifted-contour regularity: ζ′/ζ ≪ log^A on the line Re w = Re s − d under the paper’s hypotheses (Thm 7.4(iv); under RH via Lemma 3.1).
Needed to dominate R_{X,α,d}; automatic under RH for d∈(0,1/2) but hypothesized in the general localization theorem.

invented entities (2)

Gaussian–Perron prime-force defect Δ_{X,α} independent evidence
purpose: Horizontal comparison of the smoothed prime field P_{X,α} to ζ′/ζ, used as the local diagnostic object.
Defined by Eq. (5)/(31); a mathematical construction with an explicit series representation, not an external physical entity. Independent handle is the prime-side sum and numerical check in §8.
Anisotropic Gaussian damping functional Q_α(ρ′;ρ₀) independent evidence
purpose: Separates amplified vs suppressed non-selected zero contributions and supplies the finite-window certificate.
Definition 7.1; pure analytic device derived from the kernel modulus. No external ontology beyond the explicit formula.

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Pith. "Pith review of A Gaussian-Perron Prime-Side Defect and Local Profiles Near Critical-Line Zeros of the Riemann Zeta Function." pith.science (2026). https://pith.science/paper/7JHX6BBR

@misc{pith2026260704316,
  author       = {Pith},
  title        = {Pith review of: A Gaussian-Perron Prime-Side Defect and Local Profiles Near Critical-Line Zeros of the Riemann Zeta Function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7JHX6BBR}},
  note         = {Machine review of arXiv:2607.04316}
}

read the original abstract

We introduce a Gaussian--Perron prime-force defect that compares a smoothed prime-side logarithmic force with the logarithmic derivative of the Riemann zeta function. The construction turns the explicit formula into a local diagnostic for zero geometry. Its kernel produces an error-function prime weight and an anisotropic zero-side damping law, with an explicit boundary separating amplified and suppressed nonlocal zero contributions. We prove an exact zero-side formula, derive a universal selected-zero profile on the logarithmic scale, and formulate a finite-window damping certificate for non-selected residues. Under explicit damping, pole, and contour-regularity hypotheses, these ingredients localize the full defect near a selected zero. Assuming the Riemann Hypothesis and the stated pole-damping condition, the full defect near each fixed simple critical-line zero has the selected-zero profile up to an exponentially small nonlocal remainder. The framework provides a local diagnostic for zero geometry associated with the Riemann zeta function.

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Works this paper leans on

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Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.

[1] [1]

On a positivity property of the Riemannξ-function

Jeffrey C. Lagarias. “On a positivity property of the Riemannξ-function”. In:Acta Arithmetica89.3 (1999), pp. 217–234.doi:10.4064/aa-89-3-217-234

work page doi:10.4064/aa-89-3-217-234 1999

[2] [2]

A monotonicity property of Riemann’sξ function and a reformulation of the Riemann Hypothesis

Jonathan Sondow and Cristian Dumitrescu. “A monotonicity property of Riemann’sξ function and a reformulation of the Riemann Hypothesis”. In:Periodica Mathematica Hungarica60.1 (2010), pp. 37–40.doi:10.1007/s10998-010-1037-3. 24

work page doi:10.1007/s10998-010-1037-3 2010

[3] [3]

Horizontal monotonicity of the modulus of the zeta function,L-functions, and related functions

Yuri Matiyasevich, Filip Saidak, and Peter Zvengrowski. “Horizontal monotonicity of the modulus of the zeta function,L-functions, and related functions”. In:Acta Arithmetica166.2 (2014), pp. 189–200.doi:10.4064/aa166-2-4

work page doi:10.4064/aa166-2-4 2014

[4] [4]

A monotonicity property of Riemann’sξfunction along certain curves

Gajendra Singh. “A monotonicity property of Riemann’sξfunction along certain curves”. In:Indian Journal of Pure and Applied Mathematics56.1 (2025), pp. 262–265. doi:10.1007/s13226-023-00476-1

work page doi:10.1007/s13226-023-00476-1 2025

[5] [5]

On a positivity property of the real part of the logarithmic derivative of the Riemannξ-function

Edvinas Goldˇ stein and Andrius Grigutis. “On a positivity property of the real part of the logarithmic derivative of the Riemannξ-function”. In:Journal of Mathematical Inequalities18.3 (2024), pp. 829–845.doi:10.7153/jmi-2024-18-45

work page doi:10.7153/jmi-2024-18-45 2024

[6] [6]

Note on the positivity of the real part of the log-derivative of the Riemannξ-function near the critical line

Andrius Grigutis and Lukas Turˇ cinskas. “Note on the positivity of the real part of the log-derivative of the Riemannξ-function near the critical line”. In:Lithuanian Math- ematical Journal(2026). Published online 28 January 2026.doi:10.1007/s10986- 026-09708-3

work page doi:10.1007/s10986- 2026

[7] [7]

E. C. Titchmarsh and D. R. Heath-Brown.The Theory of the Riemann Zeta-Function. 2nd ed. Revised by D. R. Heath-Brown. Oxford: Clarendon Press, 1986.isbn: 0-19- 853369-1

1986

[8] [8]

Edwards.Riemann’s Zeta Function

Harold M. Edwards.Riemann’s Zeta Function. Vol. 58. Pure and Applied Mathematics. New York: Academic Press, 1974.isbn: 0-12-232750-0

1974