Definition:Riemann Zeta Function

The Riemann Zeta Function $$\zeta \left({s}\right) \ $$ is defined as

$$\zeta \left({s}\right) = \sum_{n=1}^\infty n^{-s} \ $$ for $$\Re \left({s}\right) > 1 \ $$.

The function can be extended to the complex plane $$\mathbb{C} - \left\{{1}\right\} \ $$ as

$$ \zeta \left({s}\right) = \tfrac{1}{(1-2^{1-s})} \sum_{n=1}^\infty \tfrac{(-1)^{n-1}}{n^s} \ $$.

Other equivalent extensions exist; of note is the definition

$$\zeta(z) = \prod_{p \text{ prime}} \frac{1}{1-p^{-z}} \ $$;

see Equivalency of Riemann Zeta Function Definitions.

Important Values
$$\zeta (2) = \tfrac{\pi^2}{6} \ $$

$$\zeta (1) \to +\infty \ $$

$$\zeta (-2n) = 0 \ $$ $$ ( \forall n \in \mathbb{N} ) \ $$