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Consider the equation $$1 + 2 + 3 + 4 + \cdots = - \frac{1}{12},$$ "proved" by Ramanujan Euler. One correct way to interpret this is that $\zeta(-1) = - \frac{1}{12},$ where $\zeta(s) = \sum_{n = 1}^{\...

There are ill-posed summations that we can assign values to, take for concreteness, $$ S = \sum_{k=0}^\infty k $$ to which we can assign $-1/12$ by several methods. Is there a fundamental and rigorous ...