Let $n$ be a positive integer, show that $$\sigma(n) \leq n^{3/2}+n$$
I'm not sure how to prove this. I believe i'm suppose to use this result I just proved, $n$ is composite if and only if $\sigma(n) \geq n + \sqrt{n}$ but i'm not sure how to do that. Any help is appreciated thanks!
In fact, the ratio $g(n) = \sigma(n)/n^{3/2}$ is a multiplicative function of $n$, and you should be able to check that the only prime power $p^k$ for which $g(p^k) > 1$ is $p^k=2^1$, and then further check that the only integer $n$ for which $g(n) > 1$ is $n=2$. That is more than enough to establish the desired inequality.