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Laurent Series $\sum_{n=-1}^\infty a_n x^n$ when $a_{-1} = \infty$

There is an asymptotic for sums of the form $$g(t) = \sum_{n=1}^{\infty} f(nt)$$ given by Zagier for $C^{\infty}$ functions $f$ of sufficiently rapid decay (Proposition 3 of his appendix on the Mellin …
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