[VA] SRC #012e - Then and Now: Roots

[J-F Garnier]

Here is where I am presently: I implemented the R(X) function and verified that I got the correct values for R(1000), R(1E6) ... R(1E9).
Then I explored how R(x) behaves for X<<1. It constantly decreases (staying positive) down to log(x)=-22 at least, meaning there is no root greater than 1E-10.
Then, unfortunately, my implementation is only giving numeric garbage, so I can't get any conclusion.
Some details: following Valentin's advice, I implemented the zeta function as a table with values from the literature until zeta(16), then computed the following terms.
The R(x) function is evaluated using the Horner algorithm, after identifying the highest term needed.

I'm afraid I will not be able to go further, unless some has a better idea to evaluate R(X).
Below the program I used, if this can be helpful for others.

J-F

10 ! SRC12E
20 OPTION BASE 1
30 M=250
40 DIM F(M),Z(M)
50 !
60 DATA 1.64493406685,1.20205690316,1.08232323371,1.03692775514
70 DATA 1.01734306198,1.00834927738,1.00407735620,1.00200839283
80 DATA 1.00099457513,1.00049418860,1.00024608655,1.00012271335
90 DATA 1.00006124814,1.00003058824,1.00001528226
100 Z(1)=1
110 FOR K=2 TO 16 @ READ Z(K) @ NEXT K
120 FOR K=K TO M
130 N=2 @ S=0
140 A=N^(-K) @ S=S+A
150 IF A>=1.E-13 THEN N=N+1 @ GOTO 140
160 Z(K)=1+S
170 NEXT K
180 FOR K=1 TO M @ F(K)=FACT(K) @ NEXT K
190 !
200 DEF FNR(L)
210 K=8
220 A=L^K/(K*F(K)*Z(K+1))
230 IF ABS(A)>1.E-13 THEN K=K+2 @ GOTO 220
240 R=0
250 FOR K=K TO 1 STEP -1
260 A=1/(K*F(K)*Z(K+1))
270 R=R*L+A
280 NEXT K
290 FNR=R*L+1
300 END DEF
310 !
320 DISP "R(1E3)=";FNR(LOG(1000))
330 DISP "R(1E6)=";FNR(LOG(1.E+6))
340 DISP "R(1E9)=";FNR(LOG(1.E+9))
350 DISP "R(1E12)=";FNR(LOG(1.E+12))
360 DISP "R(4E16)=";FNR(LOG(4.E+16))
370 !
380 DISP "LOG(X) R(X)"
390 FOR L=0 TO -40 STEP -1
400 DISP L;FNR(L)
410 NEXT L

>RUN

R(1E3)= 168.359446282
R(1E6)= 78527.3994306
R(1E9)= 50847455.4255
R(1E12)= 37607910540.7
R(4E16)= 1.07529277872E15

LOG(X) R(X)
0 1
-1 .557331425899
-2 .325459335158
-3 .19890131983
-4 .126895395868
-5 .08422535701
[...]
-20 .00260522574
-21 .002335956316
-22 .002172618526
-23 .002356240694
-24 .003210607624
[...]
-31 .013358513742
-32 -.75233322896
-33 -3.53759316048
-34 -10.161375979
-35 -23.8480080257
-36 -54.9637121895