Riemann's Zeta Function - another approach (RPL)

[Dieter]

(07-02-2017 02:54 PM)Dieter Wrote:  First, there is a new approximation for 0,97≤x≤1,03 with an error of approx. ±0,5 units in the 10th significant digit.

Update: I remembered the Zeta evaluation using a polynomial with the Stieltjes' constants, and so the idea was to develop a dedicated polynomial approximation. Here I chose a simple quadratic equation so that only three constants are required. This is what I came up with:

Zeta(x) ~ 1/u + 0,577215665 + 0,07281553 · u – 0,0048452 · u²
where u = x–1  and  0,965 ≤ x ≤ 1,035

The error within the given domain is less than 4 E–9, i.e. here less than 0,4 ULP (if evaluated with sufficient precision, that is).

One more term (i.e. a third order polynomial) improves the results substantially, you can easily handle 0,9 ≤ x ≤ 1,1 with a possible accuracy better than 0,2 units in the 10th significant digit:

Zeta(x) ~ 1/u + 0,5772156632 + 0,072815845 · u – 0,00484404 · u² – 0,0003424 · u³
where u = x–1 and 0,9 ≤ x ≤ 1,1

One more update:

Think big. ;-) I tried a fourth-order polynomial and the results are quite promising. You can go all the way from 1,1 down to 0,5 (!) with an error less than half a unit in the 10th significant digit. Maybe you want to try this:

Zeta(x) ~ 1/u + 0,5772156625 + 0,072815839 · u – 0,004844823 · u² – 0,000339474 · u³ + 0,000104308 · u⁴
where u = x–1 and 0,5 ≤ x ≤ 1,1

Dieter