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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.

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