Let \(\{c_ n\}\) be a sequence in the complex plane C with lim \(c_ n=\infty\). \textit{W. Luh} [Colloq. Math. Soc. János Bolyai 19, 503-511 (1978; Zbl 0411.30017)] proved the existence of an entire function F such that, for every compact set \(K\subset C\) with connected complement, and for every function f(z) that is holomorphic in the interior of K and continuous on the boundary of K, there exists a subsequence \(\{c_{n_ k}\}\) for which \(F(z+c_{n_ k})\) converges uniformly to f on K. Such a function F is called an additively universal entire function. The author considers the replacement of C by a Riemann surface S and the replacement of \(F(z+c)\) by F(g(z)), where g is an automorphism of S. It is proved that there exists a multiplicatively universal holomorphic function F on \(C^*=C-\{0\}\); i.e. one can choose \(S=C^*\) and \(g(z)=cz\). In this case \(\{c_ n\}\) is chosen to have 0 and \(\infty\) as limit points. Remarks are made concerning other choices for S and g.