No. While it's true that ζ(-1) = -1/12 and that the ζ function plays an important role in physics, your reasoning is fallacious.
If X implies Y and we know that Y is true, that does not mean X is true. So just because the video reached a "correct" conclusion does not mean that the means by which they reached that that conclusion are sensible or even consistent.
If I threw a dart at a dartboard labeled "What is 1 + 2 + 3 + ...?" and it landed on the section marked "-1/12", would you believe my answer? Would the fact that it happened to land on "-1/12" and also agreed with the ζ function lend credibility to my dart-throwing method of proof?
Indeed, if I encapsulated the methods used in that video, I could use those methods to have 1 + 2 + 3 + ... turn out to be any number I choose. This is the problem with specious reasoning — one can use it to reach any conclusion.
If X implies Y and we know that Y is true, that does not mean X is true. So just because the video reached a "correct" conclusion does not mean that the means by which they reached that that conclusion are sensible or even consistent.
If I threw a dart at a dartboard labeled "What is 1 + 2 + 3 + ...?" and it landed on the section marked "-1/12", would you believe my answer? Would the fact that it happened to land on "-1/12" and also agreed with the ζ function lend credibility to my dart-throwing method of proof?
Indeed, if I encapsulated the methods used in that video, I could use those methods to have 1 + 2 + 3 + ... turn out to be any number I choose. This is the problem with specious reasoning — one can use it to reach any conclusion.