# Zeros

## Distribution

$\displaystyle {\displaystyle Z(t) = \exp@{\iunit \vartheta(t)} \RiemannZeta@{\tfrac{1}{2}+\iunit t} }$

Substitution(s): $\displaystyle {\displaystyle {\displaystyle \vartheta(t) \equiv \ph@@{\EulerGamma@{\tfrac{1}{4} + \tfrac{1}{2}\iunit t}} - \tfrac{1}{2} t \ln@@{\cpi}}}$

Constraint(s): $\displaystyle {\displaystyle Z(t)\in\Real}$ &
${\displaystyle {\displaystyle \vartheta (t)}}$ is chosen to make ${\displaystyle {\displaystyle Z(t)}}$ real &
$\displaystyle {\displaystyle \ph@@{\EulerGamma@{\tfrac{1}{4}+\tfrac{1}{2}\iunit t}}}$ assumes its principal value

## Riemann-Siegel Formula

$\displaystyle {\displaystyle Z(t) = 2 \sum_{n=1}^m \frac{\cos@{\vartheta(t) - t \ln@@{n}}}{n^{1/2}} + R(t) }$

This formula has the name: Riemann-Siegel formula