Riemann's Zeta Function - another approach (RPL)

[Gerson W. Barbosa]

\[\zeta (s)= \frac{2^{s}}{2^{s}-2}\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(k+1)^{s}}\]

\[\zeta (s)\approx \frac{2^{s}}{2^{s}-2}\left ( \frac{1}{2\cdot \left (n+\frac{1}{2}+\frac{s+1}{8\cdot n}-\frac{2\cdot s+1}{27\cdot n^{2}} \right )^{s}} + \sum_{k=1}^{n} \frac{(-1)^{k}}{(n-k+1)^{s}} \right )\]


If I were to guess, I'd say the third term (n and 1/2 are too trivial to count) is (3*s + 1)/(64*n^3), but I haven't tried it yet.

Here is a Q&D stack-only implementation:

Code:


%%HP: T(3)A(D)F(.);
\<< DUPDUP TYPE { C\->R MIN } IFT -1.3 ^ 178. * 1. + 2. / IP DUP + DUP2 0. SWAP 2. / 1.
  FOR k OVER k DUP + DUP 1. - PICK3 ^ INV UNROT SWAP ^ INV SWAP UNROT - + -1.
  STEP 4. ROLLD DUP + 1. + OVER SQ 27. * / PICK3 1. + PICK3 8. * / SWAP - SWAP .5 + + OVER ^ DUP + INV ROT + 2. ROT ^ DUP 2. - / *
\>>

0.5 -> -1.46035450880 (6.17 s)
1.0001 -> 10000.5772771 (2.54 s)
1.27 -> 4.30022020082 (1.88 s)
1.5 -> 2.61237534865 (1.53 s)
2 -> 1.64493406684 (1.04 s)
3 -> 1.20205690315 (0.64 s)
4 -> 1.08232323371 (0.48 s)
5 -> 1.03692775514 (0.37 s)
6 -> 1.01734306199 (0.32 s)
7 -> 1.00834927738 (0.27 s)
12 -> 1.00024608656 (0.18 s)
19.99 -> 1.0000009606 (0.12 s)
(2,3) -> (0.798021985140,-0.113744308037) (4.22 s)

Edited to correct a sign in the second formula.

Update: it looks like the denominator of the second term is 24, not 27. The previous term, (s+1)/(8*n) is correct.

Now using

\[\zeta (s)\approx \frac{2^{s}}{2^{s}-2}\left ( \frac{1}{2\cdot \left (n+\frac{1}{2}+\frac{s+1}{8\cdot n}-\frac{2\cdot s+1}{24\cdot n^{2}}+\frac{3\cdot s+1}{30\cdot n^{3}} \right )^{s}} + \sum_{k=1}^{n} \frac{(-1)^{k}}{(n-k+1)^{s}} \right )\]

n should always be even.

Not sure about the last two correction terms, though. I'd need to do some calculations using 30+ SD in order to determine them.

Code:


%%HP: T(3)A(D)F(.);
\<< DUPDUP TYPE { C\->R MIN } IFT -1.27 ^ 157. * 1. + 2. / IP DUP + DUP2 0. SWAP 2. / 1.
  FOR k OVER k DUP + DUP 1. - PICK3 ^ INV UNROT SWAP ^ INV SWAP UNROT - + -1.
  STEP 4. ROLLD 3. * 1. + OVER 3. ^ 30. * / PICK3 DUP + 1. + PICK3 SQ 24. * / - PICK3 1. + PICK3 8. * / + .5 + + OVER ^ DUP + INV ROT + 2. ROT ^ DUP 2. - / *
\>>

0.5 -> -1.46035450880 (5.36) [378]
1.0001 -> 10000.5772772 (2.24) [156]
1.27 -> 4.30022020086 (1.70) [116]
1.5 -> 2.61237534869 (1.39) [94]
2 -> 1.64493406686 (0.98) [66]
3 -> 1.20205690316 (0.60) [38]
4 -> 1.08232323371 (0.44) [26]
5 -> 1.03692775514 (0.36) [20]
6 -> 1.01734306198 (0.31) [16]
7 -> 1.00834927738 (0.28) [14]
12 -> 1.00024608655 (0.17) [6]
19.99 -> 1.0000009606 (0.14) [4]
(2,3) -> (0.798021985137,-0.113744308000) (3.90) [66]

(t): time in seconds
[n]: evaluated terms in the alternating series.