$\textit{The Computations of Polylogarithms, 1992,Technical report UKC, University of Kent, Canterbury, UK}\\ $ by $\textbf{David C Wood}$ says that $$ Li_p(e^w) = \sum_{n\geq 0} \zeta(p-n)\frac{w^n}{n!}+\Gamma(1-p)(-w)^{p-1} $$ when $|w| < 2\pi$, $p\neq 1,2,3\dots$, $p\in \mathbb{R}$.
- Is there any known expression/upper bound for $$Li_p(e^w)-\Gamma(1-p)(-w)^{p-1}$$ in the case $|w| \geq 2\pi$ and p is a negative real?
- Also, I am trying to estimate the sum $$\sum_{n\geq 0} \zeta(1-\sigma-n)\frac{(-2\pi t)^n}{n!}$$ when $\sigma$ is a large positive real and $0<t<1$. Using the functional equation of Riemann zeta function, I noticed that it equals $$\sum_{n\geq 0} \cos(\pi/2(\sigma+n))\Gamma(\sigma+n)\zeta(\sigma+n)\frac{(-t)^n}{n!}$$ and that it is bounded above trivially by $\Gamma(\sigma)\zeta(\sigma,1-t)$ by Ramanaujan's formula. Does there exist such estimates for for $t\geq 1$ case? Any reference related to these quesions would also be hugely appreciated.
