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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.

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No, no science "does that": either homourists trying to ge a laugh, serious mathematicians trying to make an important point about somethibng...or, of course, cranks. –  DonAntonio Apr 29 at 20:51
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@DonAntonio: And into which of these three categories should we include Ramanujan ? :-) –  Lucian Apr 29 at 21:02
    
Most probably into all three, @Lucian...though I'd say that before he understood his great genius was an unpolished gem he was more into the ignorant category, something that has happened sometimes to several geniuses. –  DonAntonio Apr 29 at 21:05
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The statement $\sum n^2 = \zeta(-2)$ is a definition, it should not be interpreted in any other way. In other words, we are not using the definition $\sum n^2 = \lim_{N \rightarrow \infty} \sum_{n=1}^N n^2$. –  RghtHndSd Apr 29 at 21:15

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