[talanum1 (OP)]

The result of = ∞ is inconsistent with ζ(-1) = -1/12. One can write this as: not = ∞. Clearly an inconsistency! (Where [&#8734?] = ∞).

I saw a video where they happily plough with the contradiction.

[Bored chemist]

I saw a video

Have you a link?

[talanum1 (OP)]

Yes:

at timestamp: 35:28.

[talanum1 (OP)]

This must have to do with analytic continuation of ζ(s). Can you call it analytic continuation if the continuation contains an expression with a pole in it?

[talanum1 (OP)]

We must find the source of the inconsistency.

[Origin]

We must find the source of the inconsistency.

You should go ahead and find the source and then let us know.
Thanks.

[talanum1 (OP)]

The source is: ζ(s) is just defined for Re(s) > 1, so we can simply not believe: .
 ([&#8734?] = ∞)

[Origin]

The source is: ζ(s) is just defined for Re(s) > 1, so we can simply not believe: .
 ([&#8734?] = ∞)

Thanks for finding the source.  So, I guess we are done here.

[talanum1 (OP)]

We have ς (s) only defined for Re(s) > 1, and analytic continuation says: assign some values for ς(s) for Re (s) <= 1 that analytically follows from it's value for Re (s) > 1. These values are wholly determined from it's values for Re(s) > 1, so they need not be the same as expected from the definition of ς (s). We must take these values as an (inconvenient) fiction.

[Eternal Student]

Hi.

The last post seems to have the right idea (minor issues with the way you phrased it - but that's not important).

We have ς (s) only defined for Re(s) > 1

    It's only defined as  ζ(s)  =       when   Re(s) >1.         That's what I think you meant.

We must take these values as an (inconvenient) fiction.

  ?  They are something but they aren't the infinite sum of the same series described above,  if that's what you meant.

Best Wishes.