I would like to know if there is a closed form solution for
$$G(n)=\sum\limits_{d\mid n}(-1)^{\frac{n}{d}}\phi(d)$$
It seems quite likely there is since $$\sum\limits_{d\mid n}\phi(d)=n$$
But I cant see any general way to approach this nutshell. Any thoughts?
Let $n=2^km$ where $m$ is odd. For $(-1)^{\frac nd}$ to be odd, we need $d$ to be a multiple of $2^k$. So we could rewrite the sum as
$$\sum_{d\mid n}(-1)^{\frac nd}\phi(d)=\sum_{\substack{d\mid n\\2^k\nmid d}}\phi(d)-\sum_{\substack{d\mid n\\2^k\mid d}}\phi(d).$$
If $k=0$, the first sum in the RHS is empty and the result is simply $-n$. If $k\geq1$, $2^k\nmid d$ is equivalent to $d\mid\frac n2$. For the right sum we can write $d=2^ke$:
$$\sum_{d\mid n}(-1)^{\frac nd}\phi(d)=\sum_{d\mid\frac n2}\phi(d)-\sum_{e\mid m}\phi(2^ke).$$
As $e$ is odd every time, $\phi(2^ke)=2^{k-1}\phi(e)$. So $$\sum_{d\mid n}(-1)^{\frac nd}\phi(d)=\sum_{d\mid\frac n2}\phi(d)-2^{k-1}\sum_{e\mid m}\phi(e).$$
The first sum in the RHS is $\frac n2$, the second is $2^{k-1}m=\frac n2$. Hence their difference is $0$.