Riemann Zeta Function
Definition
The function defined as $\zeta : \mathbb{C} \setminus \left\{ 1 \right\} \to \mathbb{C}$ is called the Riemann Zeta Function. $$ \zeta (s) := \sum_{n \in \mathbb{N}} n^{-s} = \prod_{p : \text{prime}} \left( 1- {p^{-s}} \right)^{-1} $$
Related Theorems
[0] Ramanujan Sum: If $\displaystyle \sum_{n \in \mathbb{N}} x^{n-1} = {{ 1 } \over { 1-x }}$ is accepted to hold even at $|x| = 1$, $$ \zeta (0) = 1 + 1 + 1 + 1 + \cdots = - {{ 1 } \over { 2 }} $$
[1] Ore’s Proof: The reason why $\zeta (1)$ is undefined is as follows. $$ \zeta (1) = \sum_{n \in \mathbb{N}} {{ 1 } \over { n }} = \infty $$
[2] Euler’s Proof: $$ \zeta (2) = \sum_{n \in \mathbb{N}} {{ 1 } \over { n^{2} }} = {{ \pi^{2} } \over { 6 }} $$
[a] Relation with the Gamma Function: If $\operatorname{Re} (s) > 1$, then $$ \zeta (s) \Gamma (s) = \mathcal{M} \left[ {{ 1 } \over { e^{x} - 1 }} \right] (s) = \int_{0}^{\infty} {{ x^{s-1} } \over { e^{x} - 1 }} dx $$
[b] Relation with the Dirichlet Eta Function: $$ \eta (s) := \sum_{n \in \mathbb{N}} (-1)^{n-1} n^{-s} $$
Description
The zeta function converges for complex numbers greater than $1$ in real part, that is, for $\operatorname{Re} (s) > 1$ within $s$ and has a relation with the Gamma Function. It has been particularly of interest in number theory and complex analysis and is infamous for being the subject of the Riemann Hypothesis.