📂Functions

Analytic Proof that 1+1+1+1+1+⋯=-1/12

Theorem

$$\begin{align*} & 1 + 1 + 1 + 1 + 1 + \cdots \\ =& \sum_{n \in \mathbb{N}} {{ 1 } \over { n^{0} }} \\ =& \zeta (0) \\ =& -{{ 1 } \over { 2 }} \end{align*}$$

Explanation

If you only focus on how adding positive numbers results in a negative number, you will never understand this post. The key is that $\sum_{n \in \mathbb{N}} 1$ is expressed as the Dirichlet series $\displaystyle \sum_{n \in \mathbb{N}} {{ 1 } \over { n^{0} }}$, and it is calculated as the function value $\zeta (0)$ of its analytic continuation, the Riemann Zeta Function $\zeta$. If you are going to show an attitude like “Anyway, the equation does not hold, does it?” or “This can show a contradiction, right?” without even trying to understand the proof properly and only taking parts that you can easily handle, it’s worse than not knowing. To be precise, this post introduces a proof solely about the equation $$\zeta (0) = - {{ 1 } \over { 2 }}$$.

Proof¹

Since the Riemann Zeta Function is analytic from $\mathbb{C} \setminus \left\{ 1 \right\}$, it is continuous from $s = 0$ and therefore $$\zeta (0) = \lim_{s \to 0} \zeta (s)$$

Riemann’s Functional Equation: $$\zeta (s) = 2^{s} \pi^{s - 1} \sin \left( {{ \pi s } \over { 2 }} \right) \Gamma (1-s) \zeta (1-s)$$

The function value of the Gamma Function $\Gamma (1-z)$ at $z=0$ is $\Gamma (1) = 0! = 1$, so when $s \to 0$, $$2^{s} \to 1 \\ \pi^{s-1} \to {{ 1 } \over { \pi }} \\ \Gamma (1-s) \to 1$$ Meanwhile, the limit at $\displaystyle \lim_{s \to 0}$ $$\sin \left( {{ \pi s } \over { 2 }} \right) \to 0 \\ \zeta (1-s) \to \infty$$ will be canceled.

Taylor Expansion of the Sine Function: $$\sin x=\sum _{ n=0 }^{ \infty }{ \frac { { x } ^{ 2n+1 } }{ (2n+1)! }{ { (-1) }^{ n } } }$$

Laurent Series of the Riemann Zeta Function: $$\zeta (s) = {{ 1 } \over { s-1 }} + \sum_{n=0}^{\infty} \gamma_{n} {{ (1-s)^{n} } \over { n! }} \qquad , s 1$$

Here $\gamma_{n}$ is defined as the $n$th Stieltjes constant as follows.

$$\gamma_{n} := \lim_{m \to \infty} \sum_{k=1}^{m} \left( {{ \left( \log k \right)^{n} } \over { k }} - {{ \left( \log m \right)^{n} } \over { n+1 }} \right)$$

Following the Taylor expansion of $\displaystyle \sin \left( {{ \pi s } \over { 2 }} \right)$ and the Laurent series of $\zeta (1-s)$,

$$\begin{align*} \lim_{s \to 0} \zeta (s) =& \lim_{s \to 0} 2^{s} \pi^{s - 1} \sin \left( {{ \pi s } \over { 2 }} \right) \Gamma (1-s) \zeta (1-s) \\ =& \lim_{s \to 0} 1 \cdot {{ 1 } \over { \pi }} \sin \left( {{ \pi s } \over { 2 }} \right) \cdot 1 \cdot \zeta (1-s) \\ =& {{ 1 } \over { \pi }} \lim_{s \to 0} \sin \left( {{ \pi s } \over { 2 }} \right) \zeta (1-s) \\ =& {{ 1 } \over { \pi }} \lim_{s \to 0} \left[ {{ \pi s } \over { 2 }} - {{ \pi^{3} s^{3} } \over { 48 }} + \cdots \right] \left[ -{{ 1 } \over { s }} + \sum_{n=0}^{\infty} \gamma_{n} {{ s^{n} } \over { n! }} \right] \\ =& {{ 1 } \over { \pi }} \lim_{s \to 0} \left[ - {{ \pi } \over { 2 }} + O (s) \right] \\ =& - {{ 1 } \over { 2 }} \end{align*}$$

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