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Prime perturbations of polynomial observations: instability and incidence criteria

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorymath.PR — Probability

Contributed by Brice Pouly ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

The cost of changing prime signs depends on how the resulting multiplicative function is observed. On a fixed prime support of finite harmonic sum, we prove that existence of a polynomial mean for every sign assignment is equivalent to uniform prime truncation and uniform convergence to local products. The criterion applies separately to ordinary and logarithmic means. For independent activations with finite expected harmonic cost, the almost-sure truncation defect is the averaged large-prime parity probability, not a sum of incidences. These criteria separate averaging methods and observation geometries. One common support is logarithmically stable for every fixed polynomial and every assignment, yet a fixed perturbation of (−1)v2(n)(-1)^{v_2(n)} on that support has divergent ordinary means on n2+1,n2+2,n2−2n^2+1,n^2+2,n^2-2. Collective upper sieves give stability at critical counting exponents without critical summability, and harmonic stability on rectangles whose side lengths have asymptotically equal logarithms. For widths XαX^\alpha, Gaussian prime estimates give the matching universal budget threshold when 119/500<α<1119/500<\alpha<1; the negative result concerns only the sum of two squares. Around every completely multiplicative sign function, fixed perturbations of arbitrarily small ∑p−σ\sum p^{-\sigma} cost can have divergent logarithmic means on n2+1n^2+1 for 125/164<σ≤1125/164<\sigma\leq1. A positive oscillation bound is common to this range, although the starting scales may depend on σ\sigma. Window laws further distinguish all origins from almost all origins.

Provenance statement

The following declaration describes the research and the published source edition: Research, mathematical exploration, review processing, source comparison, proof integration, bibliographic checks, writing and preparation of the files were assisted by GPT-6 Astra. Brice Pouly takes responsibility for the manuscript. The accompanying finite diagnostics check selected identities and examples; they do not establish the asymptotic reservoirs, imported analytic estimates, category statements or almost-sure limits. No proof-assistant formalization or independent human certification is claimed. For this Hexagon deposit, OpenAI Codex assisted source packaging, compilation checks and the preparation of submission metadata. This preparation does not constitute independent mathematical peer review. The published edition used for this deposit, including its supplementary source material, is available at https://doi.org/10.5281/zenodo.23018445. At the time of preparation for Hexagon, the submitter declares the text primarily AI-generated and declares that no part of the contribution has been checked and understood by a human.

References

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Version history

  1. v1Submitted by Brice PoulyInitial depositCurrentOct 18, 2026