(Note: this question was posted also in MSE)

I'd like to know if there's a closed formula or at least an estimate for the following (finite) sum: $$ \sum_{D|p-1} \varphi(D) \,\varphi\left(\frac{p-1}{D} \right) \frac{1}{D} $$ where $\varphi(n)$ is the Euler totient function. An upper bound would come from $$ \varphi(D) \,\varphi\left(\frac{p-1}{D} \right) \leq \varphi(p-1) $$ so I get $$ \varphi(p-1)\sum_{D|p-1}\frac{1}{D} = \varphi(p-1)\frac{\sigma(p-1)}{p-1} $$

Can I do better than this? No closed formula?