[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