Taking r>0, let π2r(x) denote the number of prime pairs (p,p+2r) with p≤x. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that π2r(x)∼2C2rli2(x) with an explicit constant C2r>0. There seems to be no good conjecture for the remainders \om_{2r}(x)=\pi_{2r(x)-2C_{2r}\,{\rm li}_2(x) that corresponds to Riemann's formula for π(x)−li(x). However, there is a heuristic approximate formula for averages of the remainders \om_{2r}(x) which is supported by numerical results