Arcs of Π(1+z^p): prime races, the Gaussian core, Dirichlet L-functions and a certified solver for Subset Sum
Source: hexagonmath.org/abs/2611.00068v1
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…
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