{"id":"hexagon/2611.00068","pdf":"https://hexagonmath.org/pdf/2611.00068v1","url":"https://hexagonmath.org/abs/2611.00068v1","tags":["cs.DS","cs.CC","math.PR","math.CO","msc:11N05","msc:11N13","msc:11M26","msc:68Q25","msc:68W20","Dirichlet L-functions"],"score":0,"title":"Arcs of Π(1+z^p): prime races, the Gaussian core, Dirichlet L-functions and a certified solver for Subset Sum","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Antonio Mennillo"],"reviews":0,"subject":"math.NT","summary":"The subsets of $\\{1\\} \\cup \\{p \\le N\\}$ reach every integer between 0 and their total, by Bertrand's postulate, so deciding whether a target is a subset sum of the primes is trivial. Counting the solutions is not: the count carries the arithmetic of the primes. We study the generating function $F_N(z) = \\Pi(1 + z^w)$ on the unit circle and follow one object, the arc at a rational $a/q$, through three settings. For the primes, the centre of the arc $a/q$ rises above the sea of the minor arcs exactly when $q = 2^k (k \\ge 2)$ or $q = 2p^j$, with excess $\\Lambda(q/2)/\\varphi(q)$, and the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Lean 4 (proof assistant)","Mathlib (proof assistant library)","Arb / python-flint (certified computation)","mpmath (certified computation)"],"discussed":0,"published":"2026-11-04","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}