Denoting the order of $g$ by $o(g)$, you can show that for any decreasing function $f$ the following inequality holds
$$\sum_{g\in G}f(o(g))\geq \sum_{g\in \mathbb Z/n\mathbb Z}f(o(g)).$$
This is because one can actually construct a bijection $\sigma:G\to\mathbb Z/n\mathbb Z$ which satisfies $$o(\sigma(g))\geq o(g)$$ for all $g\in G$. The main ingredient is a classical <a href="http://www.pitt.edu/~gmc/ch1/node7.html">theorem of Frobenius</a> saying that when $k$ divides the order of a group, the number of elements of order dividing $k$ is divisible by $k$, then proceed by induction. An application of this exact idea is for example <a href="http://www.jstor.org/stable/2695368">problem 10775</a> on the American Math Monthly. For your question we just need $f(x)=-\log x$.