Riemann's Zeta Function - another approach (RPL)

[Dieter]

(07-17-2017 05:49 PM)Gerson W. Barbosa Wrote:  Yes, but I wasn't able to get more consistent terms from that model. This appears to be more promising. But I'll look into that later.

OK. In the meantime I have set up another approximation for 1,1 ≤ x ≤ 2 that complements the recent one for smaller x. The error is less than one unit in the 12th significant digit, even with rounded coefficients. Finally I integrated all three methods in one HP41 program:

For 0 ≤ x ≤ 1,1 the already known approximation above is used.
For 1,1 < x ≤ 2 the mentioned new approximation above is applied.
For x > 2 your initial method was implemented.
Here the worst case is 22 terms or about 17 seconds execution time.
The first two methods finish in about 4...5 seconds.

This may be slightly optimized with two modified polynomial approximations that split the domain at x=1. Actually the error for 1<x<1,1 when using the x>1,1 approximation is only 1...2 units in the 12th place which does not matter much on a 10-digit calculator. ;-)

Edit: [x] done. The last program version uses two new polynomial approximations for 0≤x<1 and 1<x≤2.

Dieter