Deduce the existence of constants $C_1' > C_2' > 0$ and $s_0 > 1$ such that, for all $k\in\mathbb{N}^*$ and all $s\geq s_0$, $$C_2' s\,|P_k\cap T| \leq |P_k(s)| \leq C_1' s\,|P_k\cap T|.$$
Deduce the existence of constants $C_1' > C_2' > 0$ and $s_0 > 1$ such that, for all $k\in\mathbb{N}^*$ and all $s\geq s_0$,
$$C_2' s\,|P_k\cap T| \leq |P_k(s)| \leq C_1' s\,|P_k\cap T|.$$