Definition:Completed Riemann Zeta Function

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

$\ds \forall s \in \C: \map \xi x := \begin{cases} \dfrac 1 2 s \paren {s - 1} \pi^{-s/2} \map \Gamma {\dfrac s 2} \map \zeta s & : \map \Re s > 0 \\ \map \xi {1 - s} & : \map \Re s \le 0 \end{cases}$

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

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

  • The factor of $\dfrac 1 2$ 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.