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I need a closed form for $$ \sum_{n=1}^\infty x^{-\frac{2\pi}{n}} e^{-2\pi n}$$ where $x\in[1,\infty)$

For $x=1$ we have the sum as $$ \sum_{n=1}^\infty e^{-2\pi n}=\frac{1}{e^{2\pi}-1}$$

For $1<x<\infty$ we can write the sum as $$ \sum_{n=1}^\infty (x^{\frac{1}{n}} e^n)^{-2 \pi} $$ Any help would be appreciated. Thanks.

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  • $\begingroup$ This seems like it can be answered using a Fourier series. I am not sure though. $\endgroup$ Mar 29, 2023 at 21:27
  • $\begingroup$ @KamalSaleh Ohk. Thank you. Please give some idea so that how to use fourier series in this question. $\endgroup$
    – Max
    Mar 29, 2023 at 21:29
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    $\begingroup$ Because of the power $1/n$, I doubt that there is a closed form. $\endgroup$
    – Gary
    Mar 29, 2023 at 23:46
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    $\begingroup$ I agree with @gary and bet there's no closed form. In a pinch you can get an approximation by replacing the sum with an integral: $\int_0^\infty\, x^{-2\pi/\xi}e^{-2\pi\xi}\, d\xi = 2\sqrt{\log x}K_1(4\pi\sqrt{\log{x}})$. $\endgroup$ Mar 30, 2023 at 0:21

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