I am interested in the asymptotic expansion in $t$($t>0$) when $t\to 0^+$ of the following series
$$
\sum_{k\ge 0}e^{-k^{2/n}t}
$$
for integer $n>2$ (n=1 follows from Poisson summation formula and n=2 is trivial). Especially the second term in the expansion.

It seems the first term in the expansion is of order $O(t^{-n/2})$, this can be obtained from estimating the sum above by the corresponding integral. But the second term seems to be much subtler, I have been trying contour integral with function $z^{n-1}e^{-z^2 t}\pi\cot(\pi z^n)$ without success.

Any references or ideas are appreciated.

Edit: I am looking at the sum since I am essentially interested in the behavior of $$ \frac{d}{dt}\left[t^{n/2}\sum_{k\ge 0}e^{-k^{2/n}t}\right] $$

when $t\to 0$. especially if it is bounded at $t=0$. I thought that if I know the second term in the expansion of $\sum_{k\ge 0}e^{-k^{2/n}t}$, then I would know the derivative.

By the comments so far I only know the coefficient in front of $t^{-n/2}$, it is good to know that(by doing contour integral I also get that), but it does not seem to tell for example if it contains a $t^{-(n-1)/2}$ as the next term in the expansion.

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