  Let $k\ge 2$ be a fixed integer. We consider sums of type
$\sum_{n_1^2+\cdots+ n_k^2\le x} F(n_1,\ldots,n_k)$, taken over the
$k$-dimensional spherical region $\{(n_1,\ldots,n_k)\in {\Bbb Z}^k:
n_1^2+\cdots+ n_k^2\le x\}$, where $F:{\Bbb Z}^k\to {\Bbb C}$ is a given
function. In particular, we deduce asymptotic formulas with remainder terms for
the spherical summations $\sum_{n_1^2+\cdots+ n_k^2\le x} f((n_1,\ldots,n_k))$
and $\sum_{n_1^2+\cdots+ n_k^2\le x} f([n_1,\ldots,n_k])$, involving the GCD
and LCM of the integers $n_1,\ldots,n_k$, where $f:{\Bbb N}\to {\Bbb C}$
belongs to certain classes of functions.
