Real positive zeros of the polylogarithm function

Question

The polylogarithm function $Li_{s}(z)$ is defined as:

\begin{equation} Li_{s}(z)=\sum_{k=1}^{\infty} \dfrac{z^{k}}{k^{s}}. \end{equation}

My question is: do there exist any real positive zeros of $Li_{s}(-z)$ for positive integer $s$? I am specifically interested in the case $Li_{3}(-z)$, but of course any generalisations would be useful.

Answer 1

The answer is negative and follows from induction on $s$.

Claim: for any $s\in\mathbb{N}^+$ and $x\in\mathbb{R}^+$, we have $\operatorname{Li}_s(-x)< 0.$

The claim is trivial for $\operatorname{Li}_1(-x)=-\log(1+x)$. Now we simply consider that $\operatorname{Li}_s(0)=0$ and:

$$\frac{d}{dx}\operatorname{Li}_{s+1}(-x) = \frac{1}{x}\,\operatorname{Li}_s(-x)\leq 0, $$ from which it follows that $\operatorname{Li}_{s+1}(-x)$ is decreasing, hence negative, on $\mathbb{R}^+$.