Zeros

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Distribution

${\displaystyle{\displaystyle{\displaystyle Z(t)=\exp\left(\mathrm{i}\vartheta(t)\right)\RiemannZeta@{\tfrac{1}{2}+\mathrm{i}t}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle{\displaystyle\vartheta(t)\equiv\ph@@{\Gamma\left(\tfrac{1}{4}+\tfrac{1}{2}\mathrm{i}t\right)}-\tfrac{1}{2}t\ln\pi}}}}$

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

Riemann-Siegel Formula

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

This formula has the name: Riemann-Siegel formula

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Zeros

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