Riemann's Zeta Function - another approach (RPL)

[Gerson W. Barbosa]

(07-09-2017 04:41 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!

As already mentioned, Excel's accuracy is a limiting factor here, as well as a really precise Zeta reference function. But at least I tried one more term, i.e. a sixth order approximation. This allows an error of about 5 units in the 12th significant digit.

Zeta(x) ~ 1/u + 0,577215664856 + 0,072815841255 u – 0,004845236518 u^2 – 0,000342577549 u^3 + 0,000096239872 u^4 – 0,000007417326 u^5 – 0,00000082183 u^6
where u = x – 1  and  0 ≤ x ≤ 1,05.

Evaluated with sufficient precision, the error is approx. 5 units in the 12th significant digit.
If my Zeta function is reasonably accurate, that is. ;-)

It surely is. Testing on the HP-75C:

Code:


10 INPUT X
15 U=X-1
20 Z=U*(U*(U*(-.00000082183*U-.000007417326)+.000096239872)-.000342577549)
25 Z=(U*(U*(U*(Z-.004845236518)+.072815841255)+.577215664856)+1)/U
30 DISP Z

0.0 -> -.5
0.1 -> -.603375198(48) [56]
0.2 -> -.73392092489(1) [6]
0.3 -> -.90455925725(3) [3]
0.4 -> -1.1347977838(4) [7]
0.5 -> -1.460354508(76) [81]
0.6 -> -1.9526614482(0) [2]
0.7 -> -2.7783884455(8) [5]
0.8 -> -4.4375384159(3) [0]
0.9 -> -9.430114019(36) [40]
0.95 -> -19.4264371969
1.00 -> 9.99999999999E499
1.05 -> 20.58084430(16) [20]

Manual copying, no doublecheck (actually no check at all). Hopefully no typos.

Thanks again!

Gerson.

Edited to fix a quite visible typo (two missing digits in the result for 0.5).