Exact Formula for the Prime Active-Zone Error in the Gauss Circle Problem

We derive an exact closed-form expression for the per-s error e (s, x) when s is prime, in the (s, t) -decomposition of the primitive lattice point counting function. The formula e (s, x) = (1+T) /2−s (1+T) /2 follows from the observation that no multiple of a prime slies in the relevant counting interval. We prove that the fractional parts s (1+T) /2 are equidistributed on 0, 1) for prime s, via Vaughan’s identity and Van der Corput’s estimate applied to the phase 1/2 √ (2x−s²). This yields the exact mean E[e = T/2 ≈0. 2975, the sign rate Pr (e>0) ≈80%, and their asymptotic stability, all confirmed numerically to x= 10²7 (over 4 ×10¹1 primes).