Averaged Form of the Hardy-Littlewood Conjecture

We study the prime pair counting functions ${\pi }_{2k}(x),$ and their averages over $2k\mathrm{.}$ 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 $2k\le x^{\theta },$ $\theta >7\mathrm{/}12,$ and give an almost sharp lower bound for fairly short averages over $k\le C\log x,$ $C

1/2.$ We generalize the ideas to other related problems.