# Finding an analytic value for ζ(3)

I wanted to see if I could find an exact value for $\zeta (3)$ ($\zeta$ – being the Riemann zeta function). Euler proved in 1741 that $\zeta (2) = \frac{\pi^2}{6}$ (see the Basel Problem for more information on this topic). It has also been proved that $\zeta (4) = \frac{\pi^4}{90}$, but there exists no analytic value for $\zeta (3)$. Given that $\zeta (2) = \frac{\pi^2}{6}$ and $\zeta (4) = \frac{\pi^4}{90}$, I made the naïve assumption that $\zeta (3) = \frac{\pi^3}{n}$, where $n$ is an integer in the interval [5; 90]. Following this assumption, I made a program in Java for testing my hypothesis (it’s source code can be found here). Here’s a rundown of how the program works:

• Calculates pi to n digits.
• Approximates the Riemann zeta function to $n$ digits precision.
• The most significant digit $d$ affected, given by the amount of partial sums $r$, is: $d = log_{10}(r^s)$ (s denotes which zeta-function it is – $\zeta(s)$).
• Thus, the amount of rounds $r$ required for $n$ digits precision is thus: $r = \sqrt[s]{10^d}$.
• This also means computing $n$ digits has a computational complexity of $O(\sqrt[s]{10^n})$ (which is unsatifactory and unfeasible for large $n$).
• Finds the ordered pair (k, n) which has the smallest deviation of $x$ from $\zeta (3)$, where $x = \frac{\pi^k}{n}$.

And it works! For $\zeta(2)$ it finds $\frac{\pi^2}{6}$, and for $\zeta(4)$ it finds $\frac{\pi^4}{90}$. However, it doesn’t find anything as pleasing with $\zeta(3)$. It’s best match was $\frac{\pi^7}{2513}$, which gives a deviation of $\approx 0.000189$ using 21 decimal places of precision. The deviation is way too high to assume this is exactly the same value, so we assume there’s no exact value for $\zeta(3)$ in the form $\frac{\pi^k}{n}$. However, I thought – why limit my program to $\zeta(3)$? So I tried it out for some different values. Following is a table containing some different exact values, I believe to be correct:

Function
$\zeta(6)$
$\zeta(8)$
$\zeta(10)$

(Believed) Value
$\frac{\pi^6}{945}$
$\frac{\pi^8}{9450}$
$\frac{\pi^{10}}{93555}$

Here’s a link to the .jar file, I used (if you want to play with it):
https://dl.dropboxusercontent.com/u/19633784/zeta_approx.jar

$SSO \Delta$