Definition:Riemann Zeta Function

Definition

 * ZetaAbs.JPG

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:


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

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 as:


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

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


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

where $\Gamma$ is the Gamma function.

Important Values

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


 * $\zeta \left({s}\right) \to +\infty$ 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 the Riemann Zeta Function.

Historical Note
When Riemann 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.