Riemann's Zeta Function - another approach (RPL)

[Dieter]

(06-27-2017 01:26 AM)Gerson W. Barbosa Wrote:  On the classic HP-15C:

Gerson, what about a program listing ?-)

Finally, here are some optimized simple approximations for 1 < x ≤ 1,01.

With u = x–1:

Zeta(x) ~ 1/u + u/13,7433 + 0,57721576
(error less than ±0,5 units in the 10th significant digit)

Zeta(x) ~ 1/u + u/(u + 13,73234) + 0,577215656
(error less than ±0,5 units in the 11th significant digit)

Zeta(x) ~ 1/u + u/(u + 13,73234) + 0,577215651
(error between 0 and –1 unit in the 11th significant digit)

Zeta(x) ~ 1/u + u/(0,9 · u + 13,733437) + 0,5772156664
(error less than ±1 unit in the 12th significant digit)

The mentioned error bounds assume exact evaluation, i.e. with more digits than the target accuracy. Otherwise the resulting errors may be slightly larger.

I tried the last approximation on the 35s and indeed in the results I got only the last digit was off by one here and there. Which does not mean that larger errors may occur due to roundoff errors in intermediate results.

Dieter