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Polylogarithms: How to prove the asympotic expression $ z \le \mathrm{Li}_{s}(z) \le z(1+2z 2^{-s}), \;z<-1, \;s \gg \log_2|z|$
For $|z| < 1, s > 0$ the polylogarithm has the power series
$$\mathrm{Li}_{s}(z) = \sum_{k=1}^\infty {z^k \over k^s} = z + {z^2 \over 2^s} + {z^3 \over 3^s} + \cdots = z\left(1+ {z \over 2^s} + {...
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