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Averaged Form of the Hardy-Littlewood Conjecture

We study the prime pair counting functions $π_{2k}(x),$ 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 $π_{2k}(x)$ over $2k \leq x^θ,$ $θ> 7/12,$ 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.