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−s^2). 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= 10^27 (over 4 ×10^11 primes).