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Is it possible to give a closed form for the integral

$$ \int_{0}^{\infty} \frac{dt}{t^{3}}\rho (t)e^{-\frac{x}{n^{2}t^{2}}}$$

where $ \rho (t) = t- \left\lfloor t\right\rfloor $ is the fractional part?

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    $\begingroup$ What is the $n^2$ good for? This term can be eaten by $x$, in other words, you may assume $n=1$. Also, your integral in the question is quite different from the one in the title. $\endgroup$ Oct 10, 2013 at 20:33
  • $\begingroup$ The closest to a "closed" form that I can give is $\frac{n\sqrt{ \pi}}{2\sqrt{x}}-\frac{n^2}{2 x}\sum_{m=1}^{\infty}\left(1-e^{-x/( n^2 m^2)}\right)$ $\endgroup$ Oct 20, 2013 at 20:14

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