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Correlation laws and identifiability of multiplicative functions on quadratic traces

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

We study which completely multiplicative sign functions can be identified from the statistics of h(n2+1)h(n^2+1), h(n2+2)h(n^2+2) and h(n2−2)h(n^2-2). In a class of finite products of quadratic characters of odd prime conductor, completed without zeros and modified at primes with summable inverse square roots, diagonal correlations of orders two and four determine the function. The second-order data recover the conductor and prime corrections, leaving exactly the ramified local signs undetermined. Fourth-order detectors recover those signs, including at the small primes; the inverse also extends to quadratic components at two. No bounded horizon suffices uniformly: for every prescribed window, two finite-conductor sources can share its complete law and disagree with density arbitrarily close to one half on each trace. For an arbitrary prescribed source hh, we construct factors fg=hfg=h such that, for each factor separately, the three traces jointly have the uniform word law at every horizon. The choice is continuous and computable relative to hh, with a finite bound uniform in hh for approximation of a canonical synchronous coupling. The same choice treats every horizon sequence o(log⁡log⁡N)o(\log\log N). The factors need not be independent. Their synchronous empirical law projects exactly to the product's empirical word law; when a limiting canonical coupling exists, it retains the product's word law, which need not identify hh as a function.

Provenance statement

The following declaration describes the research and the published source edition: GPT-6 Astra assisted the exploration, writing, finite computations, reference checking and code preparation of the source notes and initial manuscript. GPT-6 Astra Pro assisted subsequent revisions, bibliographical comparisons and reproducibility checks. The author retains responsibility for the manuscript and any submission. These AI-assisted readings and finite checks are neither independent human peer review nor formal verification. 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.23318918. 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.

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Version history

  1. v1Submitted by Brice PoulyInitial depositCurrentOct 18, 2026