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The problem

Can be $\zeta(3)$ written as $\alpha\pi^\beta$, where ($\alpha,\beta \in \mathbb{C}$), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$, for example)?

Details

Several $\zeta$ values are connected with $\pi$, like:

$\zeta$(2)=$\pi^2/6$

$\zeta$(4)=$\pi^4/90$

$\zeta$(6)=$\pi^6/945$

...

and so on for all even numbers.

See this mathworld link to more details: Riemann Zeta Function

So the question is, could $\zeta(3)$ be written as:

$\zeta(3)=\alpha\pi^\beta$ $\alpha,\beta \in \mathbb{C}$ $\beta \ne 0$ $\alpha \text{ not dependent of } \pi$

See $\alpha$ not essencially belongs $\mathbb{Q}$ and $\alpha,\beta$ could be real numbers too.

When I wrote $\alpha$ is not dependent of $\pi$ it's a strange and a hard thing to be defined, but maybe $\alpha$ can be written using $e$ or $\gamma$ or $\sqrt2$ or some other constant.

Edit:

Maybe this still a open question. If

$ \sum_{k = 0}^{2} (-1)^{k} \frac{B_{2k} \ B_{2- 2k + 2}}{(2k)! \ (2 - 2k + 2)!}$

in $-4 \sum_{k = 0}^{2} (-1)^{k} \frac{B_{2k} \ B_{2- 2k + 2}}{(2k)! \ (2 - 2k + 2)!}\pi^3$ be of the form $\frac{\delta}{\pi^3}$ with $\delta$ not dependent of $\pi$

and $- 2 \sum_{k \geq 1} \frac{k^{-3}}{e^{2 \pi k} - 1}$ not dependent of $\pi$ too, this question still hard and open.

  • 0
    Maybe would intersting extend the definition not only to $\alpha\pi^\beta$ but to something like to a sum of powers in $\pi$ (a finite sum), not necessarly with integer powers, and a constant not dependent of $\pi$. But I think this will complicate so much the text and I think too the spirt of the question is clear.2012-04-12

6 Answers 6

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The question is whether or not $\zeta(3)$ is connected with $\pi$. The answer is yes. Moreover, $\zeta(3) = \beta \pi^{\alpha}$ for some complex $\alpha, \beta$. Take $\alpha = 3$ and $\beta = 0.0387682...$. It is not known whether $\beta = 0.0387682...$ is algebraic or transcendental. It is known, however, that $\zeta(3)$ is irrational as shown by Apery.

Ramanujan conjectured and Grosswald proved that the following holds. If $\alpha, \beta > 0$ such that $\alpha \beta = \pi^{2}$, then for each non-negative integer $n$, \begin{align} \alpha^{-n} \left( \frac{\zeta(2n+1)}{2} + \sum_{k \geq 1} \frac{k^{-2n-1}}{e^{2 k \alpha} - 1} \right) & = (- \beta)^{-n} \left( \frac{\zeta(2n+1)}{2} + \sum_{k \geq 1} \frac{k^{-2n-1}}{e^{2 k \beta} - 1} \right) - \end{align} \begin{align} \qquad 2^{2n} \sum_{k = 0}^{n+1} (-1)^{k} \frac{B_{2k} \ B_{2n- 2k + 2}}{(2k)! \ (2n - 2k + 2)!} \alpha^{n - k + 1} \beta^{k}. \end{align} where $B_n$ is the $n^{\text{th}}$-Bernoulli number.

For odd positive integer $n$, we take $\alpha = \beta = \pi$, \begin{align} \zeta(2n+1) = -2^{2n} \left( \sum_{k = 0}^{n+1} (-1)^{k} \frac{B_{2k} \ B_{2n- 2k + 2}}{(2k)! \ (2n - 2k + 2)!} \right) \pi^{2n+1} - 2 \sum_{k \geq 1} \frac{k^{-2n-1}}{e^{2 \pi k} - 1}. \end{align}

In particular, for $n = 1$, \begin{align} \zeta(3) = -4 \left( \sum_{k = 0}^{2} (-1)^{k} \frac{B_{2k} \ B_{2- 2k + 2}}{(2k)! \ (2 - 2k + 2)!} \right) \pi^{3} - 2 \sum_{k \geq 1} \frac{k^{-3}}{e^{2 \pi k} - 1}. \end{align}

Observe that the coefficient of $\pi^{3}$ is rational, however, nothing is known about the algebraic nature of the infinite sum. This is a current topic of research. Indeed, it is conjectured that $\frac{\zeta(3)}{\pi^{3}}$ is transcendental.

Update: Recently, Takaaki Musha claims to have proved that $\frac{\zeta(2n+1)}{(2 \pi)^{2n+1}}$ is irrational for positive $n \geq 1$. However, some objection has since been raised (read comments below).

  • 0
    The **user02138** answer is the best suggestion to solve this question. But, maybe the sum multiplying $\pi^3$ (see his answer) have a value like $\frac{\delta}{\pi^3}$, where $\delta$ can belongs to irrationals but not *dependent of $\pi$*, in this case. If it ocurrs, still a hard and open question^^.2012-04-12
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Any complex number can be written as $\alpha \pi^\beta$ if you make no assumptions on $\alpha$ and $\beta$. You can even take $\beta = 0$.

It is however conjectured that $\zeta(2n+1)/\pi^{2n+1}$ is irrational for all $n \geq 1$. This conjecture is wide open.

  • 8
    @Jason: Dear Jason, There are specific reasons, related to the geometry behind the period integrals that compute these zeta values, for conjecturing this irrationality. So it is *not* just conjectured on the basis of a genericity argument. In fact, the algebraic relations between the zeta values (or rather, the absence of such --- see my comments below the question) should reflect the highly structured geometry (the geometry of mixed Tate motives) that underlies the period integrals giving rise to the zeta values. Best wishes,2011-04-28
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See this question at MO. The short answer is that we don't know whether $\zeta(3)/\pi^3$ is rational (I assume that that's what you meant), but nobody seriously believes it is. Indeed, it shouldn't be, instead it should be connected to a certain higher regulator (google for Borel regulator and/or Lichtenbaum conjecture if you want to know more, but beware that it's pretty technical stuff).

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I'll answer your modified question, as explained in your comment:

I am not searching for a rational number, maybe could be a irrational...a strange irrational. Maybe α uses γ (constant).

It's known through LLL testing that there is no simple form of $\zeta(3)$ involving low powers of $\pi$, certain other constants like $\gamma,$ and simple (small denominator) rationals. It's not expected that such a form exists at all, but of course this is not known (and probably won't be any time soon).

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Other than the answers presented in this post, I would like to mention several formulae expressing $\zeta(3)$ (and other odd zeta values) in terms of powers of $\pi$. The most well-known ones are due to Plouffe and Borwein & Bradley:

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

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

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I'm hoping it'll turn out to be related somehow to OEIS A104007

$\zeta(3)=c \cdot \pi^3/60$

$\zeta(5)=c \cdot \pi^5/378$

$\zeta(7)=c \cdot \pi^7/2700$

where $c$ is either a constant or a function of $n$. The sequence 6,60,90,378,945,2700 are the denominators of coefficients in the expansion of $\frac{x^2}{(1-e^{-2x})^{2}}$.

  • 10
    Dear Mitch, Although there are not many theorems proved about $\zeta$ at odd integers, there is a well-developed theoretical picture which is expected to explain these values. Unlike in the case of positive even integers, it is not expected that the values at the positive odd integers should be related to one another, or to $\pi$. Regards,2011-04-28