I wanted to see if I could find an exact value for ( – being the Riemann zeta function). Euler proved in 1741 that (see the Basel Problem for more information on this topic). It has also been proved that , but there exists no analytic value for . Given that and , I made the naïve assumption that , where 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 digits precision.
- The most significant digit affected, given by the amount of partial sums , is: (s denotes which zeta-function it is – ).
- Thus, the amount of rounds required for digits precision is thus: .
- This also means computing digits has a computational complexity of (which is unsatifactory and unfeasible for large ).
- Finds the ordered pair (k, n) which has the smallest deviation of from , where .
And it works! For it finds , and for it finds . However, it doesn’t find anything as pleasing with . It’s best match was , which gives a deviation of 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 in the form . However, I thought – why limit my program to ? So I tried it out for some different values. Following is a table containing some different exact values, I believe to be correct:
Here’s a link to the .jar file, I used (if you want to play with it):