Watching this video on You Tube I got the impression that some sciences (in this case physics) use the analytic continuation of the Riemann zeta function without justification. Maybe this is just my interpretation of what was being said, but I will continue: For example, on logically arriving at something like $$\sum_{n=1}^\infty n^2,$$ one might naively replace this with $\zeta(-2)$, which has value $0$ under the analytic continuation of the Riemann zeta function.
My question is simply: do some sciences really do this? Surely any results obtained this way are suspect until proven by alternative methods?
In classical mathematics I've read that Euler used divergent definite integrals to obtain "correct" results - although these would of course require rigorous proof via alternative methods once found.