Definition:Riemann Zeta Function

Definition
The Riemann Zeta Function $\zeta$ is the complex function defined on the half-plane $\Re \paren s > 1$ as the series:


 * $\displaystyle \map \zeta s = \sum_{n \mathop = 1}^\infty \frac 1 {n^s}$

Analytic Continuation
By Analytic Continuations of Riemann Zeta Function, $\zeta$ has a unique analytic continuation to $\C \setminus \set 1$.

This analytic continuation is still called the Riemann zeta function and still denoted $\zeta$.

Also see

 * Riemann Zeta Function is Analytic
 * Euler Product for Riemann Zeta Function
 * Riemann Hypothesis

Special values

 * Basel Problem, for $\zeta(2)$.
 * Riemann Zeta Function at Even Integers
 * Riemann Zeta Function at Non-Positive Integers
 * Harmonic Series is Divergent: $\zeta \left({s}\right) \to +\infty$ as $s \to 1$
 * Trivial Zeroes of Riemann Zeta Function

Generalizations

 * Definition:Dirichlet Series

Historical Note
When first investigated this function, he used $s$ instead of the more typical complex variable $z$. To this day, $s = \sigma + it$ is still typically used as the complex variable in investigations of the zeta function.