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Reflected sign patterns of the Liouville function at every large total

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by Brice Pouly ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

For signs s,ts,t, let Ns,t(N)N_{s,t}(N) count the ordered representations N=n+(N−n)N=n+(N-n) with λ(n)=s\lambda(n)=s and λ(N−n)=t\lambda(N-n)=t. We prove that, for every fixed ε>0\varepsilon>0, all four counts are at least cεN1−εc_\varepsilon N^{1-\varepsilon} at every sufficiently large integer total. The constants are ineffective through Siegel's theorem. The same conclusion holds uniformly for completely multiplicative sign functions whose positive primes satisfy ∑f(p)=1p−θ≤M\sum_{f(p)=1}p^{-\theta}\le M, with fixed 0<θ<10<\theta<1 and MM. Separately, at each sufficiently large prime pp, both patterns with product (−1/p)(-1/p) have at least p/(4(log⁡p)96)p/(4(\log p)^{96}) representations. All four satisfy this stronger bound in six reduced classes modulo 2424 and outside one very sparse family of primes. The proof combines finite reflection-to-dilation propagation, a short-product spectral test, reconstruction from the units, and signed counting of prime disagreements. For the uniform extension, candidates are united before a common exceptional population is removed. None of the conclusions gives a positive-density bound or cancellation of the reflected correlation at every total.

Provenance statement

The following declaration describes the research and the published source edition: This work was developed with substantial assistance from GPT-6 Astra in exploratory derivations, proof organization, drafting, bibliographic work and targeted checks. Brice Pouly is the sole author and retains responsibility for the manuscript. 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.22959603. 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