Averaged Form of the Hardy-Littlewood Conjecture

We study the prime pair counting functions $\pi_{2k}(x),$ and their averages over $2k.$ We show that good results can be achieved with relatively little effort by considering averages. We prove an asymptotic relation for longer averages of $\pi_{2k}(x)$ over $2k \leq x^\theta,$ $\theta > 7/12,$ and give an almost sharp lower bound for fairly short averages over $k \leq C \log x,$ $C >1/2.$ We generalize the ideas to other related problems.