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We obtain asymptotic formulas with remainder terms for the hyperbolicsummations $\sum_{mn\le x} f((m,n))$ and $\sum_{mn\le x} f([m,n])$, where $f$belongs to certain classes of arithmetic functions, $(m,n)$ and $[m,n]$denoting the gcd and lcm of the integers $m,n$. In particular, we investigatethe functions $f(n)=\tau(n), \log n, \omega(n)$ and $\Omega(n)$. We also definea common generalization of the latter three functions, and prove acorresponding result.

On certain sums of arithmetic functions involving the gcd and lcm of two positive integers