Riemann's Zeta Function - another approach (RPL)

[Dieter]

(06-19-2017 10:13 PM)Gerson W. Barbosa Wrote:  Yes, I prepared a table with twelve arguments ranging from 1.5 to 25 and the respective number of terms required for the maximum possible accuracy then I simply did a curve fitting of the data on the HP-42S.

Hehe... I used a similar approach a few years ago for the continued fraction expansion of the Normal distribution CDF, where the number of required terms was computed with a simple formula. This is the preferred method for large x, and the number of terms increases rapidly as x drops below 2...1,5. That's why for smaller arguments a series expansion is the better and faster method. So in a way this is quite similar to what we are discussing here.

(06-20-2017 02:04 AM)Gerson W. Barbosa Wrote:  What about [snip code sample] ?

This can be done with two steps less: ;-)

Code:

C01 LBL C
C02 FP
C03 ENTER
C04 1/x
C05 2
C06 LastX
C07 x!
C08 -
C09 *
C10 +
C11 RTN

(06-20-2017 02:04 AM)Gerson W. Barbosa Wrote:  ...for 1 < x <= 1.00001 ?

I would do the switch at the point where both methods, the regular and the simplified one, have the same error. The simplified version should be close to 12 digits with x < 1,00001, or 10 digits for x < 1,0001.

(06-20-2017 02:04 AM)Gerson W. Barbosa Wrote:  This saves the eleven steps required by the constant on the HP-11C, for instance.

On a ten digit calculator and 1 < x < 1,0001 the result has at most five decimals. So the EM-constant is not required to have more either.

Code:

01 LBL C
02 FP
03 1/x
04 ,
05 5
06 7
07 7
08 2
09 2
10 +
11 RTN

Or has this been simplified a bit too much ?-)

Dieter