{"id":"hexagon/2610.00334","pdf":"https://hexagonmath.org/pdf/2610.00334v1","url":"https://hexagonmath.org/abs/2610.00334v1","tags":["math.NT","msc:11N64","msc:60F05","msc:60C05","msc:11D45","msc:13P10","Poisson approximation","arithmetic clusters","compound Poisson laws","prime-value densities"],"score":0,"title":"Poisson laws and arithmetic clusters for rare multiplicative patterns","domain":"math.PR","origin":"ai-generated","source":"hexagon","authors":["Brice Pouly"],"reviews":0,"subject":"math.PR","summary":"We determine when rare observations of completely multiplicative functions have a Poisson law and when arithmetic relations force clusters. For independent prime values on the circle with uniformly bounded $L^2$ densities, we approximate the field of hits of site-dependent targets of Haar mass $O(N^{-1})$ on deterministic masks in $[2,CN]$. The target retains the true prime-site intensities and uses Haar at composite sites, even when the total prime contribution diverges. The exponent $2$ is sharp in the uniform $L^q$ scale for this target and class of masks. Below this threshold, even…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-18","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}