The gist of it seems to be as follows. The Riemann zeta function, defined on some complex numbers, is the function f(x) = 1/1^x + 1/2^x + 1/3^x + ..., which is the limit, if it exists, of the sequence 1/1^x, 1/1^x + 1/2^x, 1/1^x + 1/2^x + 1/3^x, ... . The function is defined on all complex numbers whose real part is greater than 1. If it were defined on -1, then f(-1) would be 1 + 2 + 3 + ..., but it isn't defined on -1, because the sequence 1, 3, 6, ... has no limit. However, there does exist a unique analytic function g such that, for all complex numbers x, if f(x) is defined then so is g(x), and f(x) = g(x), and g is defined on -1. For reasons I don't understand, g(-1) = -1/12, and that somehow entitles one to say that 1 + 2 + 3 + ... = -1/12.

Come to think of it, I don't understand why the Riemann zeta function can't just be considered as a function on the reals, or even the rationals, and then proceed from there. Is it because in that context it does not extend to an analytic function that is defined on -1? Is that why it is necessary to invoke the complex numbers?

The "proof" I've seen is predicated not on the zeta function (which I don't really understand) but on other "proofs" that involve shady manoeuvres like

S1 = 1 - 1 + 1 - 1 + ... = 1/2 (because it yields partial sums 1 and 0 of which 1/2 is the mean lol)

Eliminate the impossible...

It's the not knowing whether a solution even exists. A problem is easier to solve if you know in advance that a solution exists. I keep hoping it's there, a hidden Faberge egg staring at me like the Cheshire cat, but the evidence to the contrary is accumulating. Failure is damaging to one's faith.

Accentuate the positive!