This one, I'm afraid, is just plain wrong.
The notation on the left: 1+2+3+.....
denotes the limit of the sequence of partial sums. It diverges.
The "proof" that you might see on youtube and elsewhere is nonsense.
If you go through the "proof" carefully you will see that it is fallacious.
Essentially what Jamon said.
No matter who you tell, in the normal sense it's blatantly wrong, because it's obviously a (unboundedly) divergent series. However this is not the purpose of the Riemann-Zeta function. Methods of analysis used in Riemann-Zeta are used to analyse the specifc case of the Dirichlet function
Well call it an
analytic continuation of the function, specifically taken over a complex function with real part strictly greater than 1.
I am only a second year student so I do not claim mastery over all of these concepts. However, the methods of analysis used in this branch of mathematics is not intended to follow the conventional means of maths in society. Analytic continuation exists for the sake of assigning values to expressions that would otherwise be obviously divergent.
If you want to claim that 1+2+... = -1/12 is false, then in a similar way ∞! = √(2π) is wrong as well. I am unsure of why you targeted simply one expression over the other.
Also, on a personal note, I feel that if anything these comments have been used to provide humour to mathematicians for years. Just saying that it's wrong is ruining all of the fun for a lot of people.
Just because the expression isn't actually written in terms of the Riemann-Zeta function, does not mean we can have a bit of fun by writing it without it.