[knight-of-lambd]

> Even more interesting... 1+2+3+4... = -1/12

What.. No. This sum diverges. It does not equal -1/12.

Now if you plug -1 into the Riemann zeta function, you get -1/12. One could interpret that to mean the sum 1 + 2 + 3 + .. can be mapped to -1/12.

But the sum has never, and will never equal -1/12. It diverges, simple as that.

[carterehsmith]

What is the argument here? There are certain ways of assigning a number to the divergent (by partial sums) sum. If they are well-defined, they all arrive at the same number. In this case, -1/12. There is nothing to argue here, that shit has been around for 200 years or more.

And, there are some results in physics that actually measure close to the number -1/12, from something that looks like a sum of 1+2+3... and that's kind of telling us that the whole construction is not just some math sleight-of-hand, it actually has some meaning in the real world.

Example: Casimir effect

So my point is... that kind of suggests the smoothness, or continuity, or differentiability, or whatever we want to call it, of the underlying function. The opposite of discrete. Is what my point was.

[twiceaday]

Your equation is only correct if you assume that the left hand side is computed using the analytic continuation of the Riemann zeta function. This assumption completely changes how the equation is interpreted and therefore verified. But you don't state this assumption so how are you surprised that people are taking the equation at face value and telling you it's wrong?

[carterehsmith]

Just to clarify, the above is not "my" equation. It is something that mathematicians (not me) figured out, repeatedly, many times over. Of course, the -1/12 number is preposterous, but it does turn out to have a physical meaning.

There are many ways of making the sum, including limit of partial-sums/summation by parts (the one we use most of the time), but there is also Abel summation, Borel summation, Ramanujan, Cesaro, and more. And frankly there is no reason to think that "summation by parts" is the "right" way. It is surely not "right" in physics. Example: why do we start summation from 1? Is the first element somehow more important than others? No, that is just our (ie human) arbitrary pick.

[no_identd]

You likely wanna watch this: https://www.youtube.com/watch?v=YuIIjLr6vUA

[kazinator]

> What is the argument here?

Essentially, it consists of text pulled from exactly the same source that you originally cited.

(Cue sound of hands washing in sink.)