(DM42) (Free42) Riemann Zeta function for complex variables
04-10-2023, 04:17 PM
Post: #1
 Gjermund Skailand Member Posts: 78 Joined: Dec 2013
(DM42) (Free42) Riemann Zeta function for complex variables
minor update
Riemann Zeta function for DM42, Free42
Real or complex input, accuracy 30-32 digits
Zeta(s)=Zeta(x+iy)
Uses reflection formula for input with real part less than 1
Note that Zeta(1) is not defined.(or infinity if you like)
Uses "Borwein method(2)"
Runs in 4stk mode, using local vaiables N,d,r,s, requires function "Gamma" included in raw file.

for real values: Accuracy of 30-32 digits.
for imaginary values y: The method is slow, since required iterations incerases with about 0.9*Y.
Might not be that important if using Free42...
If you do not require full accuracy, at line 16, change from 44 to 16 will give 11-12 correct digits
some exact values as examples for calibration
Zeta(2)= (pi^2)/6
Zeta(10)=(pi^10)/93555
Zeta(-13)= -1/12
No exact values for complex input, but compare accuracy to wolframalpha:

Zeta(2.5+i3.3), in 0.28 seconds on DM42 on USB power -->
0.8517959540094586535483446233861111 -i0.06816865625500691866043404518227872

For complex values, use e.g. http://www.wolframalpha.com Zeta function
Zeta(2.5 +i3.3) =
0.8517959540094586535483446233861100 -0.06816865625500691866043404518227873 i

Zeta(2.5+i3.3), reduced accuracy in 0.10 seconds on DM42 on USB power -->
0.85179595401 -i0.06816865626 (fix 11)

2023-04-10 skai new version
Code:
 00 { 220-Byte Prgm } 01▸LBL "Zeta" @ Riemann Zeta, complex accuracy 30-32 digits  02 FUNC 11    @ uses Borwein (2) 03 L4STK      @ local variables N,d,r,s, requires function "Gamma" 04 RAD        @ skai 2023-04-10 05 LSTO "s" 06 ENTER 07 REAL? 08 0 09 CPX? 10 RCOMPLX 11 X≠0? 12 GTO 09 13 X≠Y? 14 GTO 09 15 -0.5        @ Zeta(0)= -0.5 16 RTN 17▸LBL 09 18 ABS 19 0.9 20 × 21 IP 22 44 23 + 24 LSTO "N" 25 1 26 LSTO "r" 27 X≤? ST Z 28 GTO 05 29 R↑ 30 - 31 STO "s" 32 DUP 33 XEQ "Gamma" 34 PI 35 2 36 × 37 R↑ 38 +/- 39 Y↑X 40 STO+ ST X 41 × 42 PI 43 2 44 ÷ 45 R↑ 46 × 47 COS 48 × 49 STO "r" 50▸LBL 05 51 RCL "N" 52 1 53 + 54 LASTX 55 NEWMAT 56 EDIT 57 CLST 58▸LBL 00 59 RCL "N" 60 RCL+ ST Z 61 RCL ST Z 62 STO+ ST X 63 COMB 64 2 65 R↑ 66 Y↑X 67 LASTX 68 R↓ 69 X↑2 70 × 71 RCL "N" 72 × 73 R↑ 74 RCL+ ST L 75 ÷ 76 + 77 STOEL 78 ISG ST Y 79 NOP 80 J+ 81 FC? 77 82 GTO 00 83 RCLEL 84 EXITALL 85 X<>Y 86 LSTO "d" 87 - 88 EDIT 89 CLST 90▸LBL 02 91 X<>Y 92 SIGN 93 RCL+ ST L 94 +/- 95 X<>Y 96 RCL ST Y 97 ABS 98 RCL "s" 99 Y↑X 100 RCLEL 101 X<>Y 102 ÷ 103 0>? ST Z 104 +/- 105 + 106 I+ 107 FC? 77 108 GTO 02 109 RCLEL 110 EXITALL 111 R↓ 112 RCL÷ "d" 113 1 114 2 115 1 116 RCL- "s" 117 Y↑X 118 - 119 ÷ 120 RCL× "r" 121 END
brgds
Gjermund

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