HEXAGONbeta

Prime noise, polynomial observations and trace-preserving completion of multiplicative functions

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorymath.PR — Probabilitymath.DS — Dynamical Systems

Contributed by Brice Pouly ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

For every prescribed completely multiplicative sign function, every integer polynomial irreducible over Q\mathbb{Q} of degree at least two and eventually positive, and every prescribed prefix, we construct a completion preserving the polynomial trace and the prefix exactly, with vanishing correlations along all fixed admissible affine families. The changed primes have relative density zero, although the changed integers have asymptotic density one half and the changed primes have infinite harmonic mass. One noise trajectory has these properties almost surely at all positive real times. The construction combines a classical root-free prime reservoir with decreasing clock rates. Its probabilistic input is affine regularisation against arbitrary bounded weights fixed before the noise, uniformly in time after a vanishing delay. Marks retain their possibly non-convergent empirical distribution. For non-selective noise, the local channel acts through irreducible factors rather than output coordinates; square identities determine its invariant characters, and rational constants supply a time-evolving random translation. The proofs separate arithmetic moments, temporal concentration and finite Fourier analysis. The constant-source existence case already follows from work of Klurman-Mangerel-Teräväinen. The conclusions concern the constructed function, not new cancellation for the unperturbed source.

Provenance statement

The following declaration describes the research and the published source edition: Research and writing were assisted by GPT-6 Astra; the author retains responsibility for the manuscript. The accompanying source package provides reproducible finite diagnostics and their verification record. These diagnostics are supplementary, not premises of the asymptotic proofs or a formal certification of them. 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.22959769. 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

  1. S. Arango-Piñeros, D. Keliher and C. Keyes, Mertens' theorem for Chebotarev sets , International Journal of Number Theory 18 (2022), no. 8, 1823--1842. 10.1142/S1793042122500932. 2103.14747 .DOI
  2. M. Aymone and V. Sidoravicius, Partial sums of biased random multiplicative functions , Journal of Number Theory 172 (2017), 343--382. 10.1016/j.jnt.2016.08.020. 1412.1157 .DOI
  3. D. Berend and Y. Bilu, Polynomials with roots modulo every integer , Proceedings of the American Mathematical Society 124 (1996), no. 6, 1663--1671. 10.1090/S0002-9939-96-03210-8.DOI
  4. R. de la Bretèche, V. Y. Wang and M. W. Xu, Random Multiplicative Functions and Making Squares from Polynomial Values , preprint, 2026. 2607.06398v1 . Theorem numbering refers to this version.
  5. B. Efron and C. Stein, The jackknife estimate of variance , The Annals of Statistics 9 (1981), no. 3, 586--596. 10.1214/aos/1176345462.DOI
  6. C. Elsholtz, B. Klahn and M. Technau, On polynomials with roots modulo almost all primes , Acta Arithmetica 205 (2022), no. 3, 251--263. 10.4064/aa220407-9-7. 2206.13466v2 .DOI
  7. A. Fish, Random Liouville functions and normal sets , Acta Arithmetica 120 (2005), no. 2, 191--196. 10.4064/aa120-2-6. math/0509315 .DOI
  8. S. R. Garcia, E. S. Lee, J. Suh and J. Yu, An effective analytic formula for the number of distinct irreducible factors of a polynomial , Journal of the Australian Mathematical Society 113 (2022), no. 3, 339--356. 10.1017/S1446788721000227. 2012.03102 .DOI
  9. O. Klurman, A. P. Mangerel and J. Teräväinen, On Elliott's conjecture and applications , Proceedings of the London Mathematical Society 133 (2026), no. 3, article e70205. 10.1112/plms.70205. Theorem and corollary numbers cited here refer to 2304.05344v2 ; editorial numbering is not used.DOI
  10. C. McDiarmid, On the method of bounded differences , in J. Siemons (ed.), Surveys in Combinatorics, 1989 , London Mathematical Society Lecture Note Series 141 , Cambridge University Press, 1989, 148--188. 10.1017/CBO9781107359949.008.DOI
Show all 14 referencesShow fewer references
  1. J. Najnudel, On consecutive values of random completely multiplicative functions , Electronic Journal of Probability 25 (2020), article 59, 28 pp. 10.1214/20-EJP456. 1702.01470 .DOI
  2. R. O'Donnell, Analysis of Boolean Functions , Cambridge University Press, 2014; corrected version, 2021. 2105.10386 .
  3. J.-P. Serre, On a theorem of Jordan , Bulletin of the American Mathematical Society 40 (2003), no. 4, 429--440. 10.1090/S0273-0979-03-00992-3.DOI
  4. L. Warnke, On the method of typical bounded differences , Combinatorics, Probability and Computing 25 (2016), no. 2, 269--299. 10.1017/S0963548315000103. 1212.5796 . Only the classical bounded-differences inequality recalled in Theorem 1 is used.DOI

Version history

  1. v1Submitted by Brice PoulyInitial depositCurrentOct 18, 2026