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The series $sum_{n=1}^infty frac{1}{n^p}$ converges for $p>1$; I have known this result since I took calculus in my freshman year. It is also known that $$sum_{n=1}^infty frac{1}{n^2}=frac{pi^2}{6} text{ and } sum_{n=1}^infty frac{1}{n^4}=frac{pi^4}{90}$$

I learned a few years later--while taking a history and philosophy of mathematics course of all places--that the precise value for which the series converges for $p=3$ is still unknown. Doing some brief investigation--using Wolfram|Alpha and Wikipedia--it appears that the result is defined in terms of the Riemann zeta function, i.e. $$sum_{n=1}^infty frac{1}{n^3}=zeta(3)approx 1.2020569...$$

I have been told that a closed form has not been found. My question is why the mechanisms we have developed that enable us to find a closed form in the previous two cases fail in the third case? What it is that makes the $p=3$ case much more difficult? Is there even a closed form for the $p=3$ case, and for that matter, for all odd-numbered cases? I have attempted to ask one of my math professors this question, but I did not quite understand the explanation at the time.

For added context, I am currently doing an undergraduate degree in statistics. As such, I have only taken two semesters of real analysis, and one semester of complex analysis. I also feel like this question might have been have been asked on this site before; if so, please point me in the right direction.

Roger Apéry proved in 1978 that $zeta(3)=displaystyle sum_{n=1}^{infty} n^{-3}$ is irrational, however the irrationality of $displaystylefrac{zeta(3)}{pi^3}$ is still an open problem. No closed form for its value is currently known.

There are however some formulas expressing $zeta(3)$ (and other odd zeta values) in terms of powers of $pi$, but these are not closed forms. The most well known ones are due to Plouffe and Borwein & Bradley:

$$
begin{aligned}
zeta(3)&=frac{7pi^3}{180}-2sum_{n=1}^infty frac{1}{n^3(e^{2pi n}-1)},\
sum_{n=1}^infty frac{1}{n^3,binom {2n}n} &= -frac{4}{3},zeta(3)+frac{pisqrt{3}}{2cdot 3^2},left(zeta(2, tfrac{1}{3})-zeta(2,tfrac{2}{3}) right).
end{aligned}
$$

Moreover, in this Math.SE post we have:

$$
frac{3}{2},zeta(3) = frac{pi^3}{24}sqrt{2}-2sum_{k=1}^infty frac{1}{k^3(e^{pi ksqrt{2}}-1)}-sum_{k=1}^inftyfrac{1}{k^3(e^{2pi ksqrt{2}}-1)}.
$$

You can also check out this paper by Vepstas, which provides a nice generalization to some of these identities.

It can be shown that $frac{sin pi x}{pi x}=prod_{kge 1}(1-frac{x^2}{k^2})$. The left-hand side's Maclaurin series can be used to write $pi^{2n}$ in terms of $zeta(2k),,1le kle n$. There is no similar theoretical treatment of $zeta(m)$ for odd $mge 3$.