Riemann's Zeta Function - another approach (RPL)

[Dieter]

(07-09-2017 08:24 PM)Gerson W. Barbosa Wrote:  

(07-09-2017 04:41 PM)Dieter Wrote:  Evaluated with sufficient precision, the error is approx. 5 units in the 12th significant digit.
If my Zeta function is reasonably accurate, that is. ;-)

It surely is.

Well, I now see some errors in the 12th digit, e.g. between x=0,1 and 0,2. Here I assumed an error round 5 ULP or less, while actually it's more than that. So the approximation should get updated, some time...

(07-09-2017 08:24 PM)Gerson W. Barbosa Wrote:  Testing on the HP-75C:

0.0 -> -.5
0.1 -> -.603375198(48) [56]
0.2 -> -.73392092489(1) [6]
0.3 -> -.90455925725(3) [3]
0.4 -> -1.1347977838(4) [7]
0.5 -> -1.460354508(76) [81]
0.6 -> -1.9526614482(0) [2]
0.7 -> -2.7783884455(8) [5]
0.8 -> -4.4375384159(3) [0]
0.9 -> -9.430114019(36) [40]
0.95 -> -19.4264371969
1.00 -> 9.99999999999E499
1.05 -> 20.58084430(16) [20]

This looks quite good, mostly within the 5 ULP limit. But as already mentioned, my Zeta reference needs some improvements. For instance, it returns Zeta(0,1) = –0,603037519852, and compared with this the approximation is merely 4 ULP off, i.e. the 5 ULP limit is met. #-)

In any case the way the approximation is evaluated is absoultely crucial. Otherwise roundoff errors may spoil the result.

Update: After a quick and dirty re-evaluation of the Zeta reference, maybe you want to try this new coefficient set:

c0 =   0,577215664858
c1 =   0,07281584127
c2 = –0,004845236649
c3 = –0,000342578367
c4 =   0,000096238267
c5 = –0,000007418588
c6 = –0,000000822161

Now Zeta(0,1) should return –0,603037519853 which is only 3 ULP off.

Dieter