Definition:Completed Riemann Zeta Function

Definition

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

$\displaystyle \forall s \in \C: \xi \left({s}\right) := \begin{cases} \frac 1 2 s \left({s - 1}\right) \pi^{-s/2} \Gamma \left({\frac s 2}\right) \zeta \left({s}\right) & : \Re \left({s}\right) > 0 \\ \xi \left({1 - s}\right) & : \Re \left({s}\right) \le 0 \end{cases}$

where $\zeta \left({s}\right)$ is the Riemann zeta function.

Notes

• The function $\xi$ is often more convenient to use than the Riemann zeta function, because the factor $s \left({s - 1}\right)$ 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:

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

• None of the factors of $\xi$ except $\zeta$ has a zero in $\C \setminus \left\{{0, 1}\right\}$, so no information is lost about the non-trivial zeros.