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# Complex Zeros of the Jonquière or Polylogarithm Function

B. Fornberg and K. S. Kölbig
Mathematics of Computation
Vol. 29, No. 130 (Apr., 1975), pp. 582-599
DOI: 10.2307/2005579
Stable URL: http://www.jstor.org/stable/2005579
Page Count: 18
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## Abstract

Complex zero trajectories of the function $$F(x, s) = \sigma^\infty_{k = 1} \frac{x^k}{k^s}$$ 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 behavior for $|x| \rightarrow 1$. In particular, trajectories are found which tend towards the zeros of the Riemann zeta function $\zeta(s)$ as $x \rightarrow - 1$, and approach these zeros closely as $x \rightarrow 1 - \rho$ for small but finite $p > 0$. However, the latter trajectories appear to descend to the point $s = 1$ as $\rho \rightarrow 0$. Both, for $x \rightarrow - 1$ and $x \rightarrow 1$, there are trajectories which do not tend towards zeros of $\zeta(s)$. The asymptotic behaviour of the trajectories for $|x| \rightarrow 0$ is discussed. A conjecture of Pickard concerning the zeros of $F(x, s)$ is shown to be false.

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