Definition:Riemann Zeta Function

Definition


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


 * $\displaystyle \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


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

where $p$ ranges over the primes, 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

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

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

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

where $\Gamma$ is the Gamma function.

Important Values

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


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


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