If $n=p_1^{k_1}p_2^{k_2}\cdots p_r^{k_r}$then ,show the inequality :$ \sigma(n) \phi(n) \geq n^2(1-\frac{1}{p_1^2})(1-\frac{1}{p_2^2})\cdots(1-\frac{1}{p_r^2})$
I know the function $\sigma(n) \phi(n)$ is multiplicative hence I only have to show the inequality for $n=p^{k}$
So $\sigma(n) \phi(n)$ $\implies \frac{n(p^{k+1}-1)}{p}$ But after then how to get the desired inequality?