From a mathematical point of view, what phenomena that most likely Mathematica Wolfram encountered when calculating:
∞∑n=1(2n−1)!(2n+2)!ζ(2n)=2log(2π)−38+ζ(3)8π2
which is incorrect.
While calculating the sum from this question, I noticed that Wolfram result is containing ζ(3)8π2, which is incorrect. Although I realized that this could be a bug, I started to wonder if there are any logical explanation behind this miscalculation! Has Wolfram algorithm encountered something similar to Riemann Rearrangement Theorem?
Doing more investigations, it turns-out that Wolfram is incorrectly miscalculating the closed form of an entire class of zeta summation, except the last case which is correct.
∞∑n=1ζ(αn)(n+a)(n+b)…=∞∑n=1[Aζ(αn)n+a+Bζ(αn)n+b+…]=C+∞∑n=1ζ(αn)−1(n+a)(n+b)…=∞∑n=1[Aζ(αn)−1n+a+Bζ(αn)−1n+b+…]+C
And with the appearance of this case (the last correct closed form), I believe there is a mathematical explanation regarding a correct summation method or algorithm that gives a kind of systematic incorrect closed form if it applied in a certain way. Appreciating if someone can explore this and alert us regardless of any bug that may exist in any math app. Thanks.
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