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Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ n+h_{j}\ \text{are prime}\right\} \right|.$$ The Hardy-Littlewood $k$-tuple conjecture implies that $$\pi_{\mathcal{H}}(x)\sim C_{\mathcal{H}}\frac{x}{\log^{2}x},$$ where $C_{\mathcal{H}}$ is a constant depending on $\mathcal{H},$ and the Selberg sieve can be used to prove that $$\pi_{\mathcal{H}}(x)\ll_{\mathcal{H}}\frac{x}{\log^{2}x}.$$ Yitang Zhang recently proved that for any sufficiently large admissible set $\mathcal{H}$, we have $$\lim_{x\rightarrow\infty}\pi_{\mathcal{H}}(x)\rightarrow\infty.$$ (the current lower bound on $\mathcal{H}$ can be found here)

Question: Does Zhang's work give a lower bound on $\pi_{\mathcal{H}}$ with the correct order of magnitude? That is, can we prove that $$\pi_{\mathcal{H}}(x)\gg_{\mathcal{H}}\frac{x}{\log^{2}x}?$$

Edit: I spoke with some experts about this problem, and the answer I received was that a bound of the correct order of magnitude is likely out of reach.

Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ n+h_{j}\ \text{are prime}\right\} \right|.$$ The Hardy-Littlewood $k$-tuple conjecture implies that $$\pi_{\mathcal{H}}(x)\sim C_{\mathcal{H}}\frac{x}{\log^{2}x},$$ where $C_{\mathcal{H}}$ is a constant depending on $\mathcal{H},$ and the Selberg sieve can be used to prove that $$\pi_{\mathcal{H}}(x)\ll_{\mathcal{H}}\frac{x}{\log^{2}x}.$$ Yitang Zhang recently proved that for any sufficiently large admissible set $\mathcal{H}$, we have $$\lim_{x\rightarrow\infty}\pi_{\mathcal{H}}(x)\rightarrow\infty.$$ (the current lower bound on $\mathcal{H}$ can be found here)

Question: Does Zhang's work give a lower bound on $\pi_{\mathcal{H}}$ with the correct order of magnitude? That is, can we prove that $$\pi_{\mathcal{H}}(x)\gg_{\mathcal{H}}\frac{x}{\log^{2}x}?$$

Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ n+h_{j}\ \text{are prime}\right\} \right|.$$ The Hardy-Littlewood $k$-tuple conjecture implies that $$\pi_{\mathcal{H}}(x)\sim C_{\mathcal{H}}\frac{x}{\log^{2}x},$$ where $C_{\mathcal{H}}$ is a constant depending on $\mathcal{H},$ and the Selberg sieve can be used to prove that $$\pi_{\mathcal{H}}(x)\ll_{\mathcal{H}}\frac{x}{\log^{2}x}.$$ Yitang Zhang recently proved that for any sufficiently large admissible set $\mathcal{H}$, we have $$\lim_{x\rightarrow\infty}\pi_{\mathcal{H}}(x)\rightarrow\infty.$$ (the current lower bound on $\mathcal{H}$ can be found here)

Question: Does Zhang's work give a lower bound on $\pi_{\mathcal{H}}$ with the correct order of magnitude? That is, can we prove that $$\pi_{\mathcal{H}}(x)\gg_{\mathcal{H}}\frac{x}{\log^{2}x}?$$

Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ n+h_{j}\ \text{are prime}\right\} \right|.$$ The Hardy-Littlewood $k$-tuple conjecture implies that $$\pi_{\mathcal{H}}(x)\sim C_{\mathcal{H}}\frac{x}{\log^{2}x},$$ where $C_{\mathcal{H}}$ is a constant depending on $\mathcal{H},$ and the Selberg sieve can be used to prove that $$\pi_{\mathcal{H}}(x)\ll_{\mathcal{H}}\frac{x}{\log^{2}x}.$$ Yitang Zhang recently proved that for any sufficiently large admissible set $\mathcal{H}$, we have $$\lim_{x\rightarrow\infty}\pi_{\mathcal{H}}(x)\rightarrow\infty.$$ (the current lower bound on $\mathcal{H}$ can be found here)

Question: Does Zhang's work give a lower bound on $\pi_{\mathcal{H}}$ with the correct order of magnitude? That is, can we prove that $$\pi_{\mathcal{H}}(x)\gg_{\mathcal{H}}\frac{x}{\log^{2}x}?$$

Edit: I spoke with some experts about this problem, and the answer I received was that a bound of the correct order of magnitude is likely out of reach.

Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ n+h_{j}\ \text{are prime}\right\} \right|.$$ The Hardy-Littlewood $k$-tuple conjecture implies that $$\pi_{\mathcal{H}}(x)\sim C_{\mathcal{H}}\frac{x}{\log^{2}x},$$ where $C_{\mathcal{H}}$ is a constant depending on $\mathcal{H},$ and the Selberg sieve can be used to prove that $$\pi_{\mathcal{H}}(x)\ll_{\mathcal{H}}\frac{x}{\log^{2}x}.$$ Yitang Zhang recently proved that for any sufficiently large admissible set $\mathcal{H}$, we have $$\lim_{x\rightarrow\infty}\pi_{\mathcal{H}}(x)\rightarrow\infty.$$ (the current lower bound on $\mathcal{H}$ can be found here)

Question: Does Zhang's work give a lower bound on $\pi_{\mathcal{H}}$ with the correct order of magnitude? That is, can we prove that $$\pi_{\mathcal{H}}(x)\gg_{\mathcal{H}}\frac{x}{\log^{2}x}?$$

Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ n+h_{j}\ \text{are prime}\right\} \right|.$$ The Hardy-Littlewood $k$-tuple conjecture implies that $$\pi_{\mathcal{H}}(x)\sim C_{\mathcal{H}}\frac{x}{\log^{2}x},$$ where $C_{\mathcal{H}}$ is a constant depending on $\mathcal{H},$ and the Selberg sieve can be used to prove that $$\pi_{\mathcal{H}}(x)\ll_{\mathcal{H}}\frac{x}{\log^{2}x}.$$ Yitang Zhang recently proved that for any sufficiently large admissible set $\mathcal{H}$, we have $$\lim_{x\rightarrow\infty}\pi_{\mathcal{H}}(x)\rightarrow\infty.$$ (the current lower bound on $\mathcal{H}$ can be found here)

Question: Does Zhang's work give a lower bound on $\pi_{\mathcal{H}}$ with the correct order of magnitude? That is, can we prove that $$\pi_{\mathcal{H}}(x)\gg_{\mathcal{H}}\frac{x}{\log^{2}x}?$$