[จูกัดเหลียง]

Difficult ones, and I've just seen it.
$$\pi(x)=\sum_{1\not =k\le x}\left\lfloor\,\dfrac{\varphi(k)}{k-1}\right\rfloor $$
$$\pi(x)=\sum_{\substack{d|p_1p_2\dots p_\ell \\ \sqrt{x}\ge p_i\in\mathscr P}}\mu(d)\left\lfloor\,\dfrac{x}{d}\right\rfloor +\pi(\sqrt{x})-1$$

$$\displaystyle \sum_{p\le x}\dfrac{\chi_2(p)}{p}=\int_{2}^x\left(\dfrac{\sum_{p\le t}\dfrac{\chi_2(p)\log p}{p}}{t\log^2 t}\right) dt+O\left(\dfrac{1}{\log x}\right)$$