Let D=(\omega_n)_{n\ge0} be the multiplicative semi-group generated by the coprime integers q_1,\dotsc, q_\tau arranged in increasing order. If f is a real-valued 1-periodic function, we consider the sums S_nf(t)=\sum_{0\le k<n} f(\omega_kt). For a large class of functions, we prove the existence of a limiting variance \sigma^2 for the sequence \{S_nf/\sqrt n\}, we give a function characterization for the case when \sigma=0 and finally we prove a central limit theorem.