Where have divergent infinite sums been successfully substituted for their corresponding Riemann Zeta results? (i.e $1+2+3\dots=-1/12$)

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In this numberphile video Professor Frenkel states that divergent infinite sums have been substituted for their "finite parts" (i.e $-1/12$ for $1+2+3+\dots$) and yielded correct answers (that have then been justified in other ways). What are some examples of this?