Squarefree counting function using zeta zeros

Question

So in Wolfram alpha there's this demonstration where the zeta zeros are used to calculate the function $\sum_{n\le x} \mu^2 (n)$ :(https://demonstrations.wolfram.com/UsingZetaZerosToCountTheSquarefreeIntegers/) It seems to be pretty accurate and work out nicely, and in the description the formula that is used to calculate it shows up. It is the sum of the residues you get when using perrons formula for $\frac{\zeta(s)}{\zeta(2s)}$ , when using a rectangle as a countour, which means that the other three integrals over the three sides of the square other than the one from perrons formula, vanish. I've tried seeing this myself, and it seems to be true, I might need to check a couple details back. Thing is though, this formula doesn't show up anywhere else at all, not on Wikipedia, I don't even see any article or book with it being mentioned, so am I doing poor search or does this mean it's not known whether the three other integrals vanish?