Riemann's Zeta Function - another approach (RPL)

[Dieter]

(07-08-2017 11:30 PM)Gerson W. Barbosa Wrote:  Very nice!

http://m.wolframalpha.com/input/?i=plot+...05&x=2&y=6

This shows that the absolute error is lower for small x and higher for x closer to 1, so that in effect the maximum is roughly 1 unit in the 10th digit:

At x=0,18 the error is about 1 E–10 which is 1 ULP of the result –0,705...
At x=0,64 and x=0,9 the error is about 1 E–9 which is 1 ULP of the results –2,227... resp. –9,430...
At x=1,05 the error is about 1 E–8 which is 1 ULP of the result 20,58...

With exact coefficients and sufficient working precision this type of approximation is good for an error within 1 ULP. Since in our case the precision is limited to 10 digits the results are mostly within 1 ULP of the correctly rounded result, and 2 ULP else (especially near the error extremes). If better precision is available, the above coefficent set yields a largest error of about 1,3 ULP.

BTW the last coefficient was tweaked a bit so that Zeta(0) rounds to exactly –0,5 on a 10-digit calculator. ;-)

Update: Here is an optimized set of coefficients that returns Zeta(0) = exactly –0,5 "by design" and also limits the error to ~1,1 ULP:

c0 = +0,5772156685
c1 = +0,07281594676
c2 = –0,00484443038
c3 = –0,00034002887
c4 = +0,00010013807
c5 = –0,00000454170

(07-08-2017 11:30 PM)Gerson W. Barbosa Wrote:  Do you have something as good as that for 12-digits calculators? Thanks!

Sorry, this would require a working precision beyond what Excel has to offer. Some time ago I did something similar on the WP34s and Free42 (decimal version), which both are more capable in this regard.

Dieter