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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 ...

Consider the truncated exponential series $$P_N(z) = \sum_{n= 0}^N \frac{z^n}{n!}$$ The zeros of this series have been studied by Szëgo and others (see e.g. here). He established an asymptotic for the ...