Definition:Riemann Zeta Function

Definition

 * ZetaAbs.JPG of $\zeta$]]

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


 * $\displaystyle \zeta \left({s}\right) = \sum_{n \mathop = 1}^\infty n^{-s}$

Important Values

 * $\zeta \left({2}\right) = \dfrac{\pi^2} 6$: see Basel Problem.


 * $\zeta \left({2 n}\right) = \left({-1}\right)^{n + 1} \dfrac {B_{2n} 2^{2n - 1} \pi^{2n}} {\left({2n}\right)!}$: see Riemann Zeta Function at Even Integers.


 * $\zeta \left({s}\right) \to +\infty$ as $s \to 1$: see Sum of Reciprocals is Divergent.


 * $\zeta \left({-2 n}\right) = 0$ for all $n \in \Z_{> 0}$: see Trivial Zeroes of Riemann Zeta Function.

Also see

 * Analytic Continuation of Riemann Zeta Function


 * Equivalence of Definitions of Riemann Zeta Function


 * Trivial Zeroes of Riemann Zeta Function


 * Riemann Hypothesis


 * Definition:General Harmonic Numbers


 * Definition:Dirichlet Series, of which from Definition 1 $\zeta \left({s}\right)$ can be seen to be an example

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.