The (Hardy or Riemann-Siegel) **Z-function** for the Riemann zeta-function is a real-valued function defined in terms of the values of $\zeta(s)$ on the critical line via the formula
\[
Z(t) := e^{i \theta(t)} \zeta\left( \frac{1}{2} + it \right),
\]
where $\theta(t)$ is the **Riemann-Siegel theta function**
\[
\theta(t) := \arg \left(\Gamma\left(\frac{2it+1}{4}\right)\right) -\frac{\log\pi}{2}t.
\]
There is a bijection between zeros $t_0$ of $Z(t)$ and zeros $\frac{1}{2}+it_0$ of $\zeta(s)$.

The **Z-function** of a general L-function is a smooth real-valued function of a real variable $t$ such that
$$|Z(t)|=|L(1/2+it)|.$$
Specifically, if we write the completed L-function as $\Lambda(s)=\gamma(s)L(s),$ where $\Lambda(s)$ satisfies the functional equation
$$\Lambda(s)=\varepsilon \overline{\Lambda}(1-s),$$
then $Z(t)$ is defined by
$$ Z(t)=\overline{\varepsilon}^{1/2} \frac{\gamma(1/2+it)}{|\gamma(1/2+it)|} L(1/2+it).$$
The square root is chosen so that $Z(t)>0$ for sufficiently small $t>0$.

The multiset of zeros of $Z(t)$ matches that of $L(1/2+it)$ and $Z(t)$ changes sign at the zeros of $L(1/2+it)$ of odd multiplicity.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Stephan Ehlen on 2019-04-30 09:18:05

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**History:**(expand/hide all)

- 2019-04-30 09:18:05 by Stephan Ehlen (Reviewed)
- 2019-03-22 13:52:28 by David Farmer
- 2019-03-22 13:51:59 by David Farmer
- 2019-03-22 11:57:49 by Brian Conrey
- 2019-03-22 11:55:03 by Brian Conrey
- 2019-03-22 11:54:03 by Brian Conrey
- 2019-03-22 11:53:38 by Brian Conrey
- 2019-03-20 11:49:05 by John Voight
- 2019-03-20 11:48:54 by John Voight
- 2018-12-19 03:20:22 by Alex J. Best

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