"Magic" regulating functions for divergent series

[Swamp Thing]

This recent video on Numberphile revisits -1 / 12 after a hiatus of nearly 10 years.



One point that they make is that there are infinitely many choices of regulating function that converge directly to the correct value (e.g. -1/12) without having to throw away "infinities" or terms of order N, N^2 etc.

Q1 : Is it true that:- If we choose a regulating function and then look at the integral corresponding to the weighted sum, and if that integral taken to infinity is zero, then that regulating function is a "magic" one? (They don't say so in the video).

Q2: If the above is true, is this particular aspect really a profound advance, or are they hyping it up just a little bit for YouTube?

 

[TensorCalculus]

No Numberphile went back to $-\frac{1}{12}$ again!? Seriously?! Last time it created such a faff...

My very limited maths knowledge means that I can't help on Q1 - but for Q2 I can imagine that this is the case, if it were so profound of an advance, there should be more on the internet than a Numberphile video about it (and I can't seem to find a ton of stuff about it on the internet about it, but maybe that's just my abysmal internet searching skills coming in)

 

[Swamp Thing]

No Numberphile went back to $-\frac{1}{12}$ again!? Seriously?! Last time it created such a faff...

My very limited maths knowledge means that I can't help on Q1 - but for Q2 I can imagine that this is the case, if it were so profound of an advance, there should be more on the internet than a Numberphile video about it (and I can't seem to find a ton of stuff about it on the internet about it, but maybe that's just my abysmal internet searching skills coming in)

It's a while since I posted those questions, and it took me a minute to even understand Q1. Which could mean I haven't worded it very clearly.

Edit: I think it would be clearer to just say "if a candidate function for the regulating function integrates to zero ... "

Edit again: No, that isn't right. It's the product of the regulating function and the envelope function that defines the terms to be summed. That product has to integrate to zero from 0 to infinity. In the case of 1+2+3.. the envelope function is f(n) = n, so the product is trivially the regulating function itself.

I think.

 

[TensorCalculus]

It's a while since I posted those questions, and it took me a minute to even understand Q1. Which could mean I haven't worded it very clearly.

Edit: I think it would be clearer to just say "if a candidate function for the regulating function integrates to zero ... "

Edit again: No, that isn't right. It's the product of the regulating function and the envelope function that defines the terms to be summed. That product has to integrate to zero from 0 to infinity. In the case of 1+2+3.. the envelope function is f(n) = n, so the product is trivially the regulating function itself.

I think.

Ah yes: I just realised the date.
Either way, I had no idea that Numberphile went back to -1/12 after so many years... I don't know what to make of that haha.

Even with a clearly worded q1... doubt I would have been able to help haha :D
But it's probably fun coming back after a year and a bit and looking back on how you worded the questions and seeing how you could have done it better!