A search on wikipedia shows: $\mu(n) = \sum_{k=1,gcd(k,n)=1}^{n} e^{2\pi i \frac{k}{n}}$ But that uses complex numbers... and requires finding out the gcd...
How useful will be a method, if that could find the exact value for $\mu(n)$ function, by just knowing all the values from $\mu(1)$ till $\mu(n-1)$, which does not require factoring any integer and which uses only elementary methods?
Has this been done before? My question is more precisely:
Does any such formula exist?