Definition:Riemann Zeta Function

Definition


The Riemann Zeta Function $$\zeta: \C \to \C \ $$ is a complex function defined as:


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

It can be seen from this definition that the Riemann zeta function is a Dirichlet series.

Other equivalent definitions exist; of note is the definition


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

See Equivalency of Riemann Zeta Function Definitions.

Analytic Continuation
The Zeta function admits of a analytic continuation to the entire complex plane, with the exception of $$s=1 \ $$ where it has a simple pole with residue $$1 \ $$.

For the right half-plane, the zeta function is defined

$$\zeta(s) = \frac{1}{1-2^{1-s}} \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s} \ $$;

with the right half-plane thus defined, the extension continues to the entire plane with

$$\Gamma \left({ \frac{s}{2} }\right) \pi^{-s/2} \zeta(s) = \Gamma \left({ \frac{1-s}{2} }\right) \pi^{\tfrac{s-1}{2}} \zeta(1-s) \ $$

where $$\Gamma \ $$ is the Gamma function.

Important Values

 * $$\zeta (2) = \tfrac{\pi^2}{6} \ $$; see Basel Problem.


 * $$\zeta (1) \to +\infty \ $$; see Sum of Reciprocals is Divergent.


 * $$\zeta (-2n) = 0 \ $$ $$ ( \forall n \in \N ) \ $$; see Trivial Zeroes of the Riemann Zeta Function.