Riemann's Zeta Function - another approach (RPL)

[Dieter]

(07-31-2017 06:31 PM)Gerson W. Barbosa Wrote:  I wonder what might have caused these differences. The coefficients were copied from your HP-41C listing:

Could you check what the polynomial returns for x=0 at the point directly before LBL 98 ? Here the value should be exactly –0,07.

But... since you use the reflection formula the approximation only has to cover the range from x=0,5 to 1. This can be done with one power less:

Zeta(x) ~ 1/u + 0,577215664944 + 0,07281584734 u – 0,00484515533 u² – 0,00034213951 u³ + 0,00009740462 u⁴ – 0,000005872685 u⁵
where u = x – 1  and  0,5 ≤ x < 1.

Within the given domain this approximation has a largest error of ~0,3 units in the 12th significant digit.

If you replace the 0...1 polynomial with this one be sure to set the last coefficient (constant term) to 0,007215664944 since the missing 0,57 is added later.
On a 12-digit calculator you can even add two more digits at make it ...494433 while the u⁵ coefficient becomes ...72675. This way the error practically matches the theoretical optimum at ~0,27 ULP.

Dieter