Arithmetic sums of Liouville type over relatively prime pairs (Q2907096)

Inspired by some results from the recent book by \textit{K. S. Williams} [Number theory in the spirit of Liouville. Cambridge: Cambridge University Press (2011; Zbl 1227.11002)], the author proves by the same arguments Liouville's type identities on sums of even integer functions ranging over sets of relatively prime pairs. For example, we quote the following identity: NEWLINE\[NEWLINE\sum(f(a- b)- f(a+ b))= (f(0)- f(n))\cdot\varphi(n),NEWLINE\]NEWLINE where \(f: \mathbb Z\to\mathbb C\) is an even function, and the sum is over \(a\), \(b\) such that \(ax+ by= n\), with \(a, b, x, y\) all odd and \((a,b)= (x,y)= 1\). Here \(\varphi\) is Euler's totient. Applications for particular functions \(f\) are also pointed out.