Zeros

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Zeros

Hurwitz Zeta Function >>

Distribution

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle Z(t) = \exp@{\iunit \vartheta(t)} \RiemannZeta@{\tfrac{1}{2}+\iunit t} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\displaystyle \vartheta(t) \equiv \ph@@{\EulerGamma@{\tfrac{1}{4} + \tfrac{1}{2}\iunit t}} - \tfrac{1}{2} t \ln@@{\cpi}}}}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle Z(t)\in\Real}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \vartheta(t)}} is chosen to make Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle Z(t)}} real &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ph@@{\EulerGamma@{\tfrac{1}{4}+\tfrac{1}{2}\iunit t}}}} assumes its principal value

Riemann-Siegel Formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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

<< Asymptotic Approximations

Zeros

Hurwitz Zeta Function >>