Complex zeros of the Jonquière or polylogarithm function

Abstract:

Complex zero trajectories of the function \[ F(x,s) = \sum \limits _{k = 1}^\infty {\frac {{{x^k}}}{{{k^s}}}} \] are investigated for real x with $|x| < 1$ in the complex s-plane. It becomes apparent that there exist several classes of such trajectories, depending on their behaviour for $|x| \to 1$. 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 $|x| \to 0$ is discussed. A conjecture of Pickard concerning the zeros of $F(x,s)$ is shown to be false.