What is $\zeta(\pi)$?

I know that $\operatorname{Re}(\pi)>1$, thus $$\zeta(\pi)=\sum_{n\geq1}\frac{1}{n^\pi}$$ Yet I have no idea how to even begin evaluating this series. It's probably unrealistic to think that there even is a closed form, but math can be funny that way sometimes.

Edit: for those of you who are confused about the definition of closed form, here's an example: $\zeta(2)=\frac{\pi^2}{6}$. Note that this is purely a ration of two constants, and even though $\pi$ cannot be exactly calculated in a finite number of operations, I am still willing to consider it a closed form. Another example: $$_3F_2\biggr(1,1,\frac{3}{2};\frac{4}{3},\frac{5}{3};\frac{2}{27}\biggl)=\frac{3\pi}{5}-\frac{6\log2}{5}$$ Note that, again, there are constants which require infinite operations to compute, yet if we just see them as constants, there is a finite number of operations in the answer, which is good enough for me. It should be noted that I do not consider a decimal expansion an adequate closed form.

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    $\begingroup$ "$\zeta(\pi)$" is most likely the closest you can get to a closed form. $\endgroup$
    – Wojowu
    Oct 21, 2018 at 18:53
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    $\begingroup$ @DietrichBurde $$\zeta(2)=\frac{\pi^2}{6}$$ Is a closed form. No decimals. $\endgroup$
    – clathratus
    Oct 21, 2018 at 18:57
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    $\begingroup$ ${\displaystyle \zeta (3)={\frac {5}{2}}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n^{3}{\binom {2n}{n}}}}.}$ is closed form. No decimals. $\endgroup$ Oct 21, 2018 at 18:58
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    $\begingroup$ $\pi$ cannot be evaluated in a finite number of operations. So $\pi^2/6$ is not in a closed form. See here:"Of course there isn’t any closed-form expression for any transcendental number including 𝜋 , since it is associated with unreal number called infinity by definition" $\endgroup$ Oct 21, 2018 at 19:07
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    $\begingroup$ Is there any closed form (in terms of elementary constant) of any zeta function? Not considering the positive even integers. $\endgroup$
    – Zacky
    Oct 21, 2018 at 19:12

1 Answer 1


Number theory currently has insufficient tools do deal with this kind of problem. We don't even have a closed form for $\zeta(3)$.


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