Complex zeros of the Jonquiere or polylogarithm function

Complex zero trajectories of the function F(x, s)= Sigma /sub k=1//sup infinity / x/sup k//k/sup s/ are investigated for real x with mod x mod <1 in the complex s-plane. It becomes apparent that there exist several classes of such trajectories, depending on their behaviour for mod x mod to 1. In particular, trajectories are found which tend towards the zeros of the Riemann zeta function zeta (s) as x to -1, and approach these zeros closely as x to 1- rho for small but finite rho >0. However, the latter trajectories appear to descend to the point s=1 as rho to 0. Both, for x to -1 and x to 1, there are trajectories which do not tend towards zeros of zeta (s). The asymptotic behaviour of the trajectories for mod x mod to 0 is discussed. A conjecture of Pickard concerning the zeros of F(x, s) is shown to be false. (20 refs).