Is $\prod\limits_{p}{p^\frac{1}{p}}$ convergent?

Question

I am trying to prove or disprove if $\prod\limits_{p}{p^\frac{1}{p}}$ converges.

I have tested up to 400K and got the following value: $$\prod_{p}{p^\frac{1}{p}}=0.26431187257195837519$$

while for natural numbers: $$\prod_{n}{n^\frac{1}{n}}=\infty$$ divergent obviously. If $$S(y)=\frac{1}{\pi}\arg(\zeta(\frac{1}{2}+iy))$$ $$\lim_{n\to\infty}\frac{1}{N}\sum_{n=1}^{N}|S(y_n)|=0.264$$ Running over the $\zeta(s)$ functions zeros. according to

https://arxiv.org/pdf/1407.4358.pdf

Not sure if those are somehow related. But looks interesting.

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Aster posting saw an obvious way to prove that the first sum divergent. Logarithm just shows this as the $\sum_{p}\frac{1}{p}$ divergent.

Answer 1

No. Taking logarithms, $$\log \prod_p p^{1/p} = \sum_p \frac{\log p}{p}$$ and this sum diverges by Mertens' "first" theorem, or even just by comparison with $ \sum_p 1/p$.