What question did this study set out to answer?

This research aims to refine the Berry-Keating arithmetic correction involving Riemann zeros, addressing prefactor errors and high-temperature fit anomalies.

May 5, 2026Open Access

ECI v7-note — Empirical Berry-Keating arithmetic correction on 10⁵ Riemann zeros (v0. 3)

Key Points

This research aims to refine the Berry-Keating arithmetic correction involving Riemann zeros, addressing prefactor errors and high-temperature fit anomalies.
Incorporated updates from prior versions and literature errata to correct prefactor and refocus on high-temperature fitting anomalies.
Utilized statistical bootstrapping and jackknife methods to analyze data for anomalies in high-temperature fits.
Presents verified parameters and constants including L_eff and GUE-first-zero calibration.
A_fit remains rejected at 6-10 sigma levels across analyzed bands, emphasizing significant inconsistencies in the arithmetic correctness.
L_eff is determined to be 7.40 with an uncertainty of +/- 0.22, consistent across varied mean lengths of the verification intervals.
M2 chi^2/dof remains unchanged at 4.17, indicating stability in findings across different analyses.

Abstract

v0. 3 (2026-05-03): consolidation update incorporating (a) the v0. 2 prefactor erratum -2/ (2pi) ² -> -2/L² (lit-triangulated Berry-Keating SIAM 41 (1999) eq. 4. 20 + Bogomolny arXiv: nlin/0312061 + Conrey-Snaith arXiv: math/0509480 Theorem 4. 1), and (b) reframe of the high-T fit anomaly previously reported as an Afit ~ 1/3 plateau / 12. 7sigma T-trend signal. Bootstrap-vs-jackknife sigmaA split is ~6. 6x (jackknife biased low along the (A, L) trade-off valley) ; after bootstrap correction, A=1 is still rejected at 6-10sigma per band, but the genuine invariant is Lₑff = 7. 40 +/- 0. 22 universal across Lₘean in 9. 1, 12. 1, with Afit ~ (Lₑff/Lₘean) ² * 0. 80 = parametric trade-off signature. No new arithmetic constant claimed. The Lₑff = 7. 40 = matched-filter window saddle (geometric anchor Lgeom = 4*xₘax*log (2) = 8. 32, sympy-verified), specifically a fit-window saturation signature, NOT a physical scale. The nu = 1. 094 cosine mode is consistent with the GUE-first-zero calibration artefact already disclosed in v0. 1 plus prime-square cluster envelope of Berry-Keating arithmetic corrections. Headline M2 chi²/dof = 4. 17 unchanged across all reframings. v0. 1 (DOI 10. 5281/zenodo. 19983241) remains as published; v0. 3 supersedes it on the Berry-Keating prefactor and the high-T interpretation only.

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