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 result of $\sum_{n=1