$$\sum_{T\text{ is PN}}\sum_{p\nmid T} [pT \le n]$$

$$\sum_{T\text{ is PN}}\sum_{p\nmid T} [pT \le n] \le \sum_{T\text{ is PN}}\sum_{p} [pT \le n]$$

$$\begin{aligned} & \Theta \left( \int_1^{\sqrt n \log^k n} \sqrt{\frac{n}{p}} \ \mathrm d \pi(p) + \int_1^{\sqrt n / \log^k n} \sqrt T \ \mathrm d T \right) \\ = \ & \Theta \left( \sqrt n \int_1^{\sqrt n \log^k n} \sqrt{\frac{1}{p}} \left(\frac{\log p - 1}{\log^2 p}\right) \ \mathrm d p + \left(\frac{\sqrt n}{\log^k n}\right)^{3/2} \right) \\ = \ & \Theta \left( \sqrt n \left(\frac{1}{2}\text{Ei}\left(\frac{\log p}{2}\right) + \frac{\sqrt p}{\log p}\right) \Big\rvert_{1}^{\sqrt n \log^k n} + \left(\frac{\sqrt n}{\log^k n}\right)^{3/2} \right) \\ = \ & \Theta \left( \sqrt n \frac{\sqrt{\sqrt n \log^k n}}{\log n} + \left(\frac{\sqrt n}{\log^k n}\right)^{3/2} \right) \\ = \ & \Theta \left( \frac{n^{3/4}}{\log^{1-k/2} n} + \frac{n^{3/4}}{\log^{3k/2} n} \right) \end{aligned}$$

$$\sum_{T\text{ is PN}}\sum_{p} [pT \le n] - \sum_{T\text{ is PN}}\sum_{p\nmid T} [pT \le n] = \sum_{T\text{ is PN}}\sum_{p\mid T} [pT \le n] \le \sum_{T\text{ is PN}}\sum_{p \le \sqrt n} [pT \le n]$$

$$\begin{aligned} &\sum_{T\text{ is PN}}\sum_{p \le \sqrt n} [pT \le n] \\ = \ & \sum_{p \le \sqrt n}\sum_{T\text{ is PN}} [T \le n / p] \\ = \ & \Theta \left( \int_1^{\sqrt n} \sqrt\frac{n}{p} \ \mathrm d\pi(p) \right) \\ = \ & \Theta \left( \sqrt n \frac{n^{1/4}}{\log n} \right) \\ = \ & \Theta \left( \frac{n^{3/4}}{\log n} \right) \end{aligned}$$

$$\begin{aligned} &\sum_{T\text{ is PN}}\sum_{p\nmid T} [pT \le n] \\ \ge \ & \sum_{T\text{ is PN}}\sum_{p} [pT \le n] - \sum_{T\text{ is PN}}\sum_{p \le \sqrt n} [pT \le n] \\ = \ & \Theta \left( \frac{n^{3/4}}{\log^{3/4} n} \right) - \Theta \left( \frac{n^{3/4}}{\log n} \right) \\ = \ & \Theta \left( \frac{n^{3/4}}{\log^{3/4} n} \right) \end{aligned}$$

$$\sum_{T\text{ is PN}}\sum_{p\nmid T} [pT \le n] = \Theta \left( \frac{n^{3/4}}{\log^{3/4} n} \right) .$$