I would like to understand the asymptotic behaviour as $a \rightarrow 0$ of the function
$$ f(a) := \sum\limits_{k=2}^{\infty} e^{ - a^2 k}{k^{-3/2}} $$
More precisely, I would like to obtain an expansion in the form $$ f(a) = f(0) - O(a^{\delta}) $$ and estimate the value of $\delta >0$.
The problem is that if I use the Taylor expansion and I compute the first derivative of $f(a)$ for $a=0$ then I obtain a divergent sum in $k$, so I think I cannot use this standard approach... do you have any suggestions?
$$f(a) = \sum\limits_{k=2}^{\infty} e^{ - a^2 k}{k^{-3/2}}=\text{Li}_{\frac{3}{2}}\left(e^{-a^2}\right)-e^{-a^2}$$
Expanded as a series $$f(a)=\left(\zeta \left(\frac{3}{2}\right)-1\right)-2 \sqrt{\pi } a+ \left(1-\zeta \left(\frac{1}{2}\right)\right)a^2+O\left(a^4\right)$$ which a good approximation for $0\leq a \leq \frac 12$.