In this paper, we obtain asymptotic formulae for the summatory functions of a class of arithmetical functions. These extend, and in certain cases refine, earlier results due to \textit{S. S. Pillai} [J. Annamalai Univ. 2, 243--248 (1933; Zbl 0008.19603)] and \textit{H. G. Kopetzky} [Monatsh. Math. 84, 213--217 (1977; Zbl 0551.10034)]. Let \(f_1, f_2,\cdots,f_r\) be arbitrary nonconstant polynomials with integer coefficients and let \(g(n)\) be an arithmetical function. We investigate the function \[ \Psi^g_{k,r}(n)=\Psi^g_{k,r}(f_1,f_2,\cdots,f_r;n)= \sum_{1\leq s_i\leq n^k, 1\leq i\leq r} g((f_1(s_1),\cdots,f_r(s_r);n^k)_k). \] Here \((a_1,\cdots,a_r;n^k)_k\) is the greatest common divisor of \(a_1,\cdots,a_r\) and \(n^k\) which is a \(k\)th power. The proofs use, among other things, a theorem of E. Wirsing.