Leaning digits and algebraic numbers

I show that there is a meromorphic continuation of Sigma({log beta n} < log beta alpha) n(-s) to the whole plane if and only if beta is a Pisot number , alpha is an element of Q(beta), and either the second largest conjugate o f beta is real or the conjugate of alpha corresponding to the second larges t conjugate of beta is positive. (C) Academie des Sciences/Elsevier, Paris.