Taking $r>0$, let $\pi_{2r}(x)$ denote the number of prime pairs $(p,\,p+2r)$ with $p\le x$. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that $\pi_{2r}(x)\sim 2C_{2r}\,{\rm li}_2(x)$ with an explicit constant $C_{2r}>0$. 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 $\pi(x)-{\rm li}(x)$. However, there is a heuristic approximate formula for averages of the remainders $\om_{2r}(x)$ which is supported by numerical results.

Additional Metadata
Keywords Hardy-Littlewood conjecture, prime-pair functions, representation by repeated complex integral, zeta's complex zeros
ACM Numerical Algorithms and Problems (acm F.2.1)
MSC Goldbach-type theorems; other additive questions involving primes (msc 11P32)
THEME Software (theme 1)
Publisher A.M.S.
Journal Mathematics of Computation
Korevaar, J, & te Riele, H.J.J. (2010). Average prime-pair counting formula. Mathematics of Computation, 79(270), 1209–1229.