{"id":"hexagon/2610.00351","pdf":"https://hexagonmath.org/pdf/2610.00351v1","url":"https://hexagonmath.org/abs/2610.00351v1","tags":["math.PR","math.DS","msc:11N64","msc:11B85","msc:60G51","affine correlations","completely multiplicative functions","finite Fourier analysis","normality","polynomial traces"],"score":0,"title":"Prime noise, polynomial observations and trace-preserving completion of multiplicative functions","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Brice Pouly"],"reviews":0,"subject":"math.NT","summary":"For every prescribed completely multiplicative sign function, every integer polynomial irreducible over $\\mathbb{Q}$ 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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-20","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}