Definition:Completed Riemann Zeta Function

Definition
The completed Riemann zeta function $\xi: \C \to \C$ is defined on the complex plane $\C$ as:


 * $\displaystyle \forall s \in \C: \map \xi x := \begin{cases}

\dfrac 1 2 s \paren {s - 1} \pi^{-s/2} \map \Gamma {\dfrac s 2} \map \zeta s & : \map \Re s > 0 \\ \map \xi {1 - s} & : \map \Re s \le 0 \end{cases}$ where $\map \zeta s$ is the Riemann zeta function.

Also known as
The completed Riemann zeta function is also known as the Riemann xi function.

Also see

 * Completed Riemann Zeta Function is Entire
 * Functional Equation for Completed Riemann Zeta Function


 * The function $\xi$ is often more convenient to use than the Riemann zeta function, because the factor $s \paren {s - 1}$ removes the simple poles of $\zeta$.

Thus the theory of entire functions can be applied to $\xi$.


 * The factor of $\displaystyle \frac 1 2$ is convenient, though omitted in some sources.

By the Gamma Difference Equation it allows us to write:


 * $\map \xi s = \paren {s - 1} \pi^{-s/2} \map \Gamma {\dfrac s 2 + 1} \map \zeta s$


 * None of the factors of $\xi$ except $\zeta$ have a zero in $\C \setminus \set {0, 1}$, so no information is lost about the nontrivial zeros.