Is it known whether computing $\mu(n)$ for a given integer $n$ is as hard as factorization?
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2$\begingroup$ Wild guess: we don't know. However, realistically (I believe Terry Tao has talked about it somewhere, will try to find it later), whatever method to compute $\mu$ we will find is likely generalize to larger number fields. Gathering such information for carefully chosen number fields will give us information about factorization of $n$ in $\mathbb N$, possibly even leading us to a full factorization. So although we might not know how to reduce factorization to finding $\mu$, a known method to find $\mu$ might lead to a factorization method. $\endgroup$ – Wojowu Nov 1 '16 at 18:52

1$\begingroup$ What I was talking about in the previous comment. $\endgroup$ – Wojowu Nov 1 '16 at 19:29