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.
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