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.
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.
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.