Casimir effect
The sum of all natural numbers One may see the following mysterious identity that the sum of all natural numbers equals to -\frac{1}{12}: \begin{equation} 1 + 2 + 3 + \cdots = -\frac{1}{12}\,.\tag{1}\end{equation} The rigorous mathematical interpretation relies on the so-called Riemann zeta function \zeta(s) and its analytic continuation. This is not the focus of this post, so we just highlight the main conclusions: \begin{equation}\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}\tag{2}\end{equation} is only valid when \mathfrak{Re} (s) > 1. After analytic continuation, we can compute \zeta(-1)=-\frac{1}{12}. Therefore, mathematically, Eq.(1) should be interpreted in the language of analytic continuation. In this post, we explain Eq. (1) by applying a scheme that is commonly used in physics when treating a divergent series \sum_{n=1}^{\infty}a_n: Multiply each term a_n by some regulator such as e^{-n\epsilon} to make the series converge. Compute the r...