# Need to verify strange behavior with Product[]

I'm running 12.1.0.0 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.

• 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. Jun 24, 2020 at 8:35
• You can try to convert the product into a sum of logarithms Jun 24, 2020 at 10:03
• Duplication of mathematica.stackexchange.com/questions/220493/… Jun 24, 2020 at 11:13
• 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. Jun 24, 2020 at 14:39
• In v12.1, NProduct gives correct numeric results. Jun 24, 2020 at 14:59

This is an answer as I cannot comment yet.

Is anyone else able to replicate this issue?

Yes, I can confirm that Mathematica 12.1.0.0 on Linux and Mac gives the answer 0, same for the odd integers greater than 5.

In Mathematica 11.3.0.0 the code

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


yields

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.