With the usual notation $\{x\}=x-\lfloor x\rfloor,$ the fractional part of $x,$ we can calculate the Fourier series of the function $f(x)=\frac12-\{x\},$ a periodic function with period $1.$ The result is $$f(x)=\sum^\infty_{n=1}\frac1{\pi n}\,\sin2\pi nx,\tag{1}$$ cf. https://mathworld.wolfram.com/FourierSeriesSawtoothWave.html.
Now let's remember Mobius inversion (cf. https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula): informally, we would obtain $$\frac1\pi\,\sin2\pi x=\sum^\infty_{n=1}\frac{\mu(n)}n\,f(nx)=\sum^\infty_{n=1}\frac{\mu(n)}n\,\left(\frac12-\{nx\}\right).\tag{2}$$
I emphasized "informally", because the assumptions of the inversion formula aren't satisfied: the series in (1) is not absolutely convergent. Is (2) still true? Here's a plot of the RHS (summing up to $n=5000000$):
So it seems to be true, but how to formally justify it? Fun fact: for $x=0,$ that would mean $$\sum^\infty_{n=1}\frac{\mu(n)}n=0,\tag{3}$$ and that's true (while the RHS of (1) is $0\neq f(x)=\frac12$ for integer $x$). It is also equivalent to the prime number theorem.
Question: Is it possible to justify (2), with or without using (3)?