Let $d$ be an odd integer. Show that $$|S_{\mathrm{prim}}(d^2)| \geq d^2\, 2^{-|\mathcal{P}(d)|}.$$ Deduce that, for all $\varepsilon>0$, we have $$d^{2-\varepsilon} = \underset{d\rightarrow+\infty}{o}\left(S_{\mathrm{prim}}(d^2)\right).$$
Let $d$ be an odd integer. Show that
$$|S_{\mathrm{prim}}(d^2)| \geq d^2\, 2^{-|\mathcal{P}(d)|}.$$
Deduce that, for all $\varepsilon>0$, we have
$$d^{2-\varepsilon} = \underset{d\rightarrow+\infty}{o}\left(S_{\mathrm{prim}}(d^2)\right).$$