Riemann's Zeta Function - another approach (RPL)

[Dieter]

(06-22-2017 01:20 AM)Gerson W. Barbosa Wrote:  For 1 < x < 1.001:

I assume this is 1 < x < 1,01.

(06-22-2017 01:20 AM)Gerson W. Barbosa Wrote:  1.009999999 GSB C -> 100.5779539
1.003456789 GSB C -> 289.8632757

I think we can top this. ;-)

Edit: the following suggestions are not yet optimized and can be improved further, please see my later post in this thread.

First of all, replace the 13,73328 with 13,7418. Evaluated exactly, this should keep the error within about ±0,7 ULP of a ten-digit result.

But we can do even better:

Zeta(x)  ~  1/u + u/(u + 13,733) + 0,57721568
where u = x–1 and 1 < x ≤ 1,01

Code:

01 LBL C
02 FRAC
03 1/x
04 LastX
05 LastX
06 13,733
07 +
08 /
09 ,57721568
10 +
11 +
12 RTN

1,009999999 => 100,5779533
1,009876543 => 101,8279365
1,003456789 => 289,8632756

According to some quick-and-dirty tests, within the given domain and with exact evaluation (!) the result should have ten significant digits ±0,2 ULP.

If you prefer the result to be either correctly rounded or truncated (error always negative) replace 0,57721568 with ...66:

Zeta (1,001001001) = 999,5772895 48

1,001001001=> 999,5772896 with ...68
1,001001001=> 999,5772895 with ...66

If that's still too much, try
Zeta(x) ~ 1/u + u/(0,9 · u + 13,7335) + 0,577215668

(resp. ...666, see above)

Here the largest error should be about ±0,02 ULP. If evaluated exactly, that is. But this improvement does not always show up on a ten-digit calculator. Actually the limiting factor seems to be the built-in 1/x function with its inherent error of 0,5 ULP.

Dieter