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Complex zeros of the Jonquière or polylogarithm function

Authors: B. Fornberg and K. S. Kölbig
Journal: Math. Comp. 29 (1975), 582-599
MSC: Primary 10H05; Secondary 33A70
MathSciNet review: 0369278
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Abstract: Complex zero trajectories of the function

$\displaystyle F(x,s) = \sum\limits_{k = 1}^\infty {\frac{{{x^k}}}{{{k^s}}}} $

are investigated for real x with $ \vert x\vert < 1$ in the complex s-plane. It becomes apparent that there exist several classes of such trajectories, depending on their behaviour for $ \vert x\vert \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 $ \vert x\vert \to 0$ is discussed. A conjecture of Pickard concerning the zeros of $ F(x,s)$ is shown to be false.

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Keywords: Jonquière function, polylogarithms, Lerch's transcendent, Fermi-Dirac integrals, Bose-Einstein integrals, Riemann zeta function, Dirichlet series, complex zeros
Article copyright: © Copyright 1975 American Mathematical Society