On a gcd-sum function

Summary.

We consider generalizations and analogues of Cesàro’s formula $\sum^{n}_{i=1} f((i, n)) = \sum_{d|n} f(d)\phi(n/d)$, where (i, n) denotes the greatest common divisor of i and n and where $\phi$ is Euler’s totient function. Particular attention is paid to the unitary analogues of this formula.