Let \(L_f(s)=\sum_{n=1}^{\infty }f(n)n^{-s}\) be the generating Dirichlet series of the arithmetic function \(f\). Supposing that \(f_1, f_2, g_1, g_2\) are completely multiplicative, the authors show by multiplicativity arguments that \[ \sum_{n=1}^{\infty }\frac{(f_1*g_1)(n)(f_2*g_2)(n)}{n^s} = \frac{L_{f_1f_2}(s)L_{g_1g_2}(s)L_{f_1g_2}(s)L_{g_1f_2}(s)}{L_{f_1f_2g_1g_2}(2s)}. \] This implies various classical formulae as special cases. Particular attention is paid to the functions \(r_N(n)\), the number of representations of \(n\) as the sum of \(N\) squares, and \(r_{2,P}(n)\), the number of solutions of the equation \(x^2+Py^2=n\), together with the squares \(r_N^2(n)\) and \(r_{2,P}^2(n)\). Generating Dirichlet series of these functions are expressed in terms of \(\zeta (s)\) and Dirichlet \(L\)-functions, and the results are applied, in the usual way, to establish asymptotic formulae for the respective summatory functions.