What I'm saying is that there is no strange mathematical result whatsoever related to the sum of the positive integers, which should be the only meaning reserved to 1+2+3+...

There are stuff like this https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7..., but writing 1+2+3+... = -1/12 is purely formal and pretending to deduce it from physics is silly because you are not performing any sum at all.