Riemann's Zeta Function - another approach (RPL)

[Gerson W. Barbosa]

(07-11-2017 06:31 PM)Dieter Wrote:  

(07-08-2017 11:30 PM)Gerson W. Barbosa Wrote:  Do you have something as good as that for 12-digits calculators? Thanks!

Gerson, I did it. ;-)

As mentioned, Excel was not sufficient to handle this, so the whole math was done on a WP34s: The Zeta reference, the matrix setup and the solving process of the resulting linear equation system. This lead to a 7th order polynomial.

Congratulations for achieving such a great accomplishment using a wp34s! A really hard work.

Done in Excel with this add-on:


 1                1
 2               -0,25
 3                0,111111111111111111111
 4               -0,0625
 5                0,04
 6               -0,0277777777777777777778
 7                0,020408163265306122449
 8               -0,015625
 9                0,0123456790123456790123
10               -0,01

                  0,8108333333333333333332


But I don't think it would have helped you much. It's rather cumbersome to use. For instance, expressions like

n^2 + n + 1 + 1/(n^2 - n) - 2/(n^2 + 1)

will have to be translated as

=xlpADD(B12*B12+B12+1;xlpSUBTRACT(xlpDIVIDE(1;B12*B12-B12);xlpDIVIDE(2;B12*B12+1)))

(07-11-2017 06:31 PM)Dieter Wrote:  With full precision (16-digit) coefficents the approximation is good for an error within 0,9 units of the 13th significant digit. The rounded coefficient set below still limits the error to 1,3 units (again in the 13th significant digit) if evaluated with sufficient precision. This way also the upper limit can be extended to 1,10621229947 which is the point beyond which Zeta drops below 10.

With 12 digit precision I have not yet found a case where the result is off by more than 1 ULP. If you do, please report here.

SysRPL handles 15 digits, in case someone wants to try, but I don't think it won't be necessary. Even (old) HP would tolerate some inaccuracy in the last digit in more simple transcendental functions.

(07-11-2017 06:31 PM)Dieter Wrote:  And here it is:

Zeta(x)  ~  1/u + 0,577215664896 + 0,0728158454506 u – 0,004845180903 u^2 – 0,00034229834 u^3 ...
... + 0,000096919523 u^4 – 0,0000065523865 u^5 – 0,0000002669693 u^6 + 0,0000001418226 u^7

where u = x–1  and  0 ≤ x ≤ 1,10621229947

If you try your examples for x = 0 to 1,1 in 0,1-steps the results are dead on.
The only exception is Zeta(0,8) which is –4,43753841589 55... On my 35s this is truncated to –4,43753841589 instead of being rounded up to ...90.

I will try it in BASIC later, but at least on an HP pocket computer :-)

Gerson.