Arcs of Π(1+z^p): prime races, the Gaussian core, Dirichlet L-functions and a certified solver for Subset Sum

The subsets of {1} ∪ {p ≤ 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) = Π(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 ≥ 2) or q = 2p^j, with excess Λ(q/2)/φ(q), and the neighbourhoods of all the other arcs are bounded uniformly in q; Theorem E gives the expansion arc by arc. The height of an arc is a race of the primes modulo q with weights log|cos(πra/q)|. By Landau's theorem a one-sided bound on it is equivalent to GRH for the even characters it sees, unconditionally at q = 8, 10 and 14, where the zero-free segments it needs are certified; at q = 14 the race changes sign infinitely often. The principal arc carries ζ in the centre and width of its Gaussian.

For arbitrary weights, the arcs become the decision rule of a Subset Sum solver that reads them on the original weights and ends with a meet-in-the-middle filtered by a random residue class. Every answer it returns is verified. The dummy weights of HURS explain why the reading is exact: they blur the counts by a flat window, and the blur is invertible on counts, though not on reachable sets. On random weights of b bits the method has rigorous guarantees, in time 2^{b/2+o(b)} and, with the filter, 2^{b/3+o(b)}, resting on a local limit theorem uniform in the size of the weights and on a concentration bound for signed subset sums. Finally, the solver is read inside a general theory of constraint propagation, in which every network factors as T = T_comp ∘ T_struct.

Code, benchmarks and the Lean 4 formalization: github.com/anthem87/arcs-algo (AGPL-3.0-or-later).