Riemann's Zeta Function - another approach (RPL)

[Gerson W. Barbosa]

(07-07-2017 07:40 PM)Dieter Wrote:  Anyway, here is a final addition. It is a fifth order approximation for 0≤x≤1,05 which has been tested in Excel with every intermediate result rounded to 10 significant digits. And indeed the results seem to match those on a real 15C or 41C. If the implementation is carefully coded, cf. step 23 ff. in the listing below, the results should be within 2 units in the 10th place. Which I think is as good as it gets with ten-digit working precision.

The formula:

Zeta(x) = 1/u + 0,577215668 + 0,0728159383 · u – 0,0048444781 · u² – 0,0003401389 · u³ + 0,0001000277 · u⁴ – 4,58184E-6 · u⁵
where u = x – 1  and  0 ≤ x ≤ 1,05.

Very nice!

http://m.wolframalpha.com/input/?i=plot+...05&x=2&y=6

Do you have something as good as that for 12-digits calculators? Thanks!

Gerson.