So exactly where does it get its value from? This is the technical bit about analytic continuation. There is a way, beyond the scope of this article to extend functions of complex numbers once you know the value of the function elsewhere. I understand why the numberphile people didn't want to bring that up because it is hard to explain nontechnically and indeed to give them credit where it is due, they have a discussion about it in an adjoining video and a written bit about it now too. So basically there is a bit of maths using function of complex numbers that allow you to assign a value to a function once you know its values elsewhere. The point of the artice was that this was too hidden for my taste which is why we still get posts like the one above where they want to still claim that the sum of all the integers is -1/12. it is not. Only a function that is related to that sum has that value. Where by related I mean that function has the same value as that sum when that sum is convergent.

yep it should be less than or equal to -1.

So exactly where does it get its value from? This is the technical bit about analytic continuation. There is a way, beyond the scope of this article to extend functions of complex numbers once you know the value of the function elsewhere. I understand why the numberphile people didn't want to bring that up because it is hard to explain nontechnically and indeed to give them credit where it is due, they have a discussion about it in an adjoining video and a written bit about it now too. So basically there is a bit of maths using function of complex numbers that allow you to assign a value to a function once you know its values elsewhere. The point of the artice was that this was too hidden for my taste which is why we still get posts like the one above where they want to still claim that the sum of all the integers is -1/12. it is not. Only a function that is related to that sum has that value. Where by related I mean that function has the same value as that sum when that sum is convergent.