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Lets consider the function : $$F(x) = \sum_{n=1} (xn)^{-s_0} e^{-nx} $$

with $s_0$ a zero of the Riemann Zeta function in the critical strip.

This sum is well defined for $x \in \mathbb{R}^{+*}$. It seems then reasonnable to think that as if $s_0$ is a zero of Zeta then $F(x)$ as the property to have zero as limit in zero. We expect to have for $x \to 0$ : $$F(x) \sim x^{1-s_0} \sum_{n=1}n^{s_0-2}$$

This what I expect because if we could use the classical Poisson summation formula with $x^{-s_0} e^{-x} $, as we have $\hat{g}(t) = Fourier(|x|^{-s_0} e^{-|x|})(t) \sim_{\infty} K_1 t^{s_0-1}+ K_2 t^{s_0-2} $ we would have (considering we can remove on both side the term n=0):

$$ \sum_{n=1} (xn)^{-s_0} e^{-nx} = \frac{1}{x} \sum_{n=1} K_1 (\frac{n}{x})^{s_0-1}+ K_2 (\frac{n}{x}) t^{s_0-2}+ O(t^{s_0-3})$$

and due to fact that $1-s_0$ is also a zero of Zeta function we finally would be able to remove first term :

$$ \sum_{n=1} (xn)^{-s_0} e^{-nx} = \frac{1}{x} \sum_{n=1} K_2 (\frac{n}{x})^{s_0-2} + O(\frac{n}{x})^{s_0-3}$$

I would like to find a reference that treat this sort of limits (or to have explanation if this is false)

I would like to use Poisson Summation formula which is particularly adapted to this type of problem but $x^{-s_0} e^{-x} $ does not fullfil classical criteria of Poisson summation formula given in litterature (singular point in zero).

This question is linked to my other question:

Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?

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