I'm running on MacOS X x86 (64-bit). The expression

Product[1 - 1/n^5, {n, 2, Infinity}]

yields 0 as output, which is clearly false. Wolfram Alpha gives the correct result. In fact, it appears that

Product[1 - 1/n^k, {n, 2, Infinity}]

for any odd integer $k \ge 5$ returns 0. Replacement of Infinity with a large positive integer gives the expected result. Is anyone else able to replicate this issue? I have a strong suspicion that this is due to some problem with handling the choice of complex branch.

  • $\begingroup$ I get the same result as W|A gives with 12.1.1. I do not have 12.1.0 installed any longer, so I cannot test that version. $\endgroup$
    – Szabolcs
    Jun 24, 2020 at 8:35
  • $\begingroup$ You can try to convert the product into a sum of logarithms $\endgroup$
    – yarchik
    Jun 24, 2020 at 10:03
  • $\begingroup$ Duplication of mathematica.stackexchange.com/questions/220493/… $\endgroup$
    – user64494
    Jun 24, 2020 at 11:13
  • $\begingroup$ On 12.0 I obtain the same result as W|A, so this looks like a regression.I suggest that you report it to Wolfram Support. $\endgroup$
    – MarcoB
    Jun 24, 2020 at 14:39
  • 1
    $\begingroup$ In v12.1, NProduct gives correct numeric results. $\endgroup$
    – Bob Hanlon
    Jun 24, 2020 at 14:59

1 Answer 1


This is an answer as I cannot comment yet.

Is anyone else able to replicate this issue?

Yes, I can confirm that Mathematica on Linux and Mac gives the answer 0, same for the odd integers greater than 5.

In Mathematica the code

Product[1 - 1/n^5, {n, 2, Infinity}] // InputForm


1/(Gamma[2 + (-1)^(1/5)]*Gamma[2 - (-1)^(2/5)]*Gamma[2 + (-1)^(3/5)]*Gamma[2 - (-1)^(4/5)])

Odd integers bigger than 5 also seem to work.


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