Averaged Form of the Hardy-Littlewood Conjecture

TL;DR

This paper investigates the average behavior of prime pair counting functions, establishing asymptotic relations for longer averages and providing bounds for shorter averages, thereby advancing understanding of prime pairs.

Contribution

It introduces new asymptotic formulas for averages of prime pair counts and extends the methods to related problems, improving upon previous bounds.

Findings

Asymptotic relation for averages over $2k \,\leq\, x^\theta$ with $\theta > 7/12$

Almost sharp lower bounds for averages over $k \leq C \log x$

Generalization of methods to other prime-related problems

Abstract

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.