On Granville's Prime-Square Refinement Of The Hardy–Littlewood Prediction For Goldbach Representations

Granville (2007) proposed refining the Hardy–Littlewood prediction for the number of representations of an even integer as a sum of two primes by subtracting a term from representations n = p + q². We argue that the coefficient 4 in his formula should be 2, because the Hardy–Littlewood main term already counts prime-square pairs with weight one half. The corrected term implies a residue law with an explicit constant. We test both versions on all even n ≤ 10⁸ and on twelve windows of 2²¹ consecutive even n between 10⁷ and 10¹⁰, with block-bootstrap errors and a pre-registered test of the mod 3 prime-race term. Under the Riemann Hypothesis we also show that the prime-only sum of log p log q over p + q ≤ N has a secondary term −(4/3)N3/2, which is missing from Granville's explicit formula (5.1) and from its corrected version.