Zeta Function

[Namir]

Here is a program that calculates the Zeta function. The program prompts you for s, the argument for the function, and for the tolerance value (recommend 1e-8). The function places the function's value in the X stack register,

Code:

1    LBL "ZETA"        
2    LBL A        
3    S?        
4    PROMPT        
5    STO 0        
6    TOLER?        
7    PROMPT        
8    STO 07        # store tolerance
9    1        
10    STO 04        # n=1
11    STO 08        # n! = 1
12    2        
13    /        
14    STO 01        # OutSum = 1/2
15    LBL 00        #--------------------------- start of outer loop 
16    RCL 04        # View n
17    PSE        
18    STO 11        # I = n
19    STO* 08        # update n!
20    RCL 08        
21    STO 10        # init (n-k)!
22    1        
23    STO 03        # k = 1
24    STO 09        # k! =1
25    STO 02        # InSum = 1
26    STO 06        # CHS=1
27    LBL 01        # --------------------------- start of inner loop
28    -1        
29    STO* 06        # CHS=-CHS
30    RCL 03        
31    STO* 09        # update k!
32    RCL 11        
33    X=0?        
34    GTO 02        
35    STO/ 10        # update (n-k)!
36    1        
37    RCL 10        
38    X=0?        
39    +        
40    STO 10        
41    1        
42    STO- 11        # I = I - 1
43    LBL 02        
44    RCL 08        
45    RCL 09        
46    /        
47    RCL 10        
48    /        
49    RCL 06        
50    *        
51    RCL 03        
52    1        
53    +        
54    STO 03        # k = k + 1
55    RCL 00        
56    Y^X        
57    /        
58    STO+ 02        # InSum = InSum + ...
59    RCL 04        
60    RCL 03        
61    X<=Y?        # k<=n?
62    GTO 01        # --------------------------- End of Inner loop
63    RCL 02        
64    2        
65    RCL 04        
66    1        
67    +        
68    STO 04        # n = n + 1
69    +        
70    Y^X        
71    /        
72    STO+ 01        # OutSum = OutSum + Term
73    ABS        
74    RCL 07        
75    X<=Y?        # Tolerance <= Term?
76    GTO 00        # --------------------------- End of Outer loop
77    2        
78    1        
79    RCL 00        
80    -        
81    Y^X        
82    CHS        
83    1        
84    +        # calculate 1 - 2^(1-s)
85    1/X        
86    RCL 01        
87    *        
88    RTN

The memory map for the program is:

Code:

R00 = s
R01 = Outer Sun, Outer Sum
R02 = Inner Sum, Inner Sum
R03 = k
R04 = n
R05 = Term
R06 = CHS
R07 = Toler
R08 = n!
R09 = k!
R10 = (n-k)!
R11 = I

The function uses a fast converging series (see Wikipedia) and can give results for small arguments of s such as Zeta(1.1).

Namir

PS: I am aware that Jean-Marie Baillard has a Zeta function implemented in the HP-41C Software Library. His program is shorter than mine and uses just two registers. However, I tried to calculate Zeta(1.1) using his version and I had to stop after the HP-41CX emulator went on and on and on!