# A curious double zeta evaluation [closed]

While investigating the evaluation of a double Euler sum (a.k.a. a double zeta function),

$$\zeta(r,s)=\sum_{j=1}^\infty \sum_{k=1}^{j-1}\frac1{j^s k^r}$$

in Mathematica, I chanced upon most peculiar behavior. Observe the following attempts to evaluate $\zeta(2,3)$:

Sum[1/(j^3 k^2), {j, 1, Infinity}, {k, 1, j - 1}]
Pi^2 Zeta[3]/6 - Zeta[5]

Sum[1/(j^3 k^2), {k, 1, j - 1}] /. j -> 1
0

Sum[1/(j^3 k^2), {j, 2, Infinity}, {k, 1, j - 1}]
(Pi^2*Zeta[3] - 11*Zeta[5])/2


One could try evaluating the internal sum in terms of the generalized harmonic numbers (i.e., Sum[HarmonicNumber[j - 1, 2]/j^3, {j, 1, Infinity}]), but the discrepancy remains. (As it turns out, the second one is the correct evaluation; you can check the consistency by using NSum[] instead of Sum[].)

A similar discrepancy can be seen in the evaluation of the double zeta function at other arguments; for instance,

Sum[HarmonicNumber[j - 1, 4]/j^5, {j, 1, Infinity}]
Pi^4 Zeta[5]/90 - Zeta[9]

Sum[HarmonicNumber[j - 1, 4]/j^5, {j, 2, Infinity}]
(Pi^4 Zeta[5] + 105 Pi^2 Zeta[7] - 1143 Zeta[9])/18


What am I doing/assuming wrong?

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Perhaps I did not understand your question. But if I use Sum[1/(j^3 k^2), {j, 1, Infinity}, {k, 1, j - 1}] and wait for some time, I get as result 1/2 (\[Pi]^2 Zeta[3] - 11 Zeta[5]). –  partial81 Feb 8 at 14:19
@partial: Hmm, are you using version 9, by any chance? My school's still stuck on version 8, and that's what it's spitting out. –  user5844 Feb 8 at 14:23
I have detected no discrepancy neither in Mathematica 8 nor in 9. E.g. In ver.8 this Sum[1/(j^3 k^2), {j, 1, Infinity}, {k, 1, j - 1}] yields 1/2 (Pi^2 Zeta[3] - 11 Zeta[5]) as well as the same with Sum[ HarmonicNumber[j - 1, 2]/j^3, {j, 1, Infinity}]. –  Artes Feb 8 at 14:28
Hmm... I tried it in 7.0.1, 8.0.4, and 9.0.1--same (correct) result every time. Have you tried this in a new session? –  Oleksandr R. Feb 8 at 14:29
Yes, I did try in a new session. Maybe something was changed in version 8.0.1? I found it curious that a simple change of lower limit gave something discrepant... –  user5844 Feb 8 at 14:31