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

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