Zeros

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Zeros

Distribution

Z ⁢ ( t ) = exp ⁡ ( i ⁢ ϑ ⁢ ( t ) ) ⁢ \RiemannZeta ⁢ @ ⁢ 1 2 + i ⁢ t 𝑍 𝑡 imaginary-unit italic-ϑ 𝑡 \RiemannZeta @ 1 2 imaginary-unit 𝑡 {\displaystyle{\displaystyle{\displaystyle Z(t)=\exp\left(\mathrm{i}\vartheta(% t)\right)\RiemannZeta@{\tfrac{1}{2}+\mathrm{i}t}}}} {\displaystyle Z(t) = \exp@{\iunit \vartheta(t)} \RiemannZeta@{\tfrac{1}{2}+\iunit t} }

Substitution(s): ϑ ⁢ ( t ) ≡ \ph ⁢ @ ⁢ @ ⁢ Γ ⁡ ( 1 4 + 1 2 ⁢ i ⁢ t ) - 1 2 ⁢ t ⁢ ln ⁡ π italic-ϑ 𝑡 \ph @ @ Euler-Gamma 1 4 1 2 imaginary-unit 𝑡 1 2 𝑡 {\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): Z ⁢ ( t ) ∈ \Real 𝑍 𝑡 \Real {\displaystyle{\displaystyle{\displaystyle Z(t)\in\Real}}} &
ϑ ⁢ ( t ) italic-ϑ 𝑡 {\displaystyle{\displaystyle{\displaystyle\vartheta(t)}}} is chosen to make Z ⁢ ( t ) 𝑍 𝑡 {\displaystyle{\displaystyle{\displaystyle Z(t)}}} real &
\ph ⁢ @ ⁢ @ ⁢ Γ ⁡ ( 1 4 + 1 2 ⁢ i ⁢ t ) \ph @ @ Euler-Gamma 1 4 1 2 imaginary-unit 𝑡 {\displaystyle{\displaystyle{\displaystyle\ph@@{\Gamma\left(\tfrac{1}{4}+% \tfrac{1}{2}\mathrm{i}t\right)}}}} assumes its principal value


Riemann-Siegel Formula

Z ⁢ ( t ) = 2 ⁢ ∑ n = 1 m cos ⁡ ( ϑ ⁢ ( t ) - t ⁢ ln ⁡ n ) n 1 / 2 + R ⁢ ( t ) 𝑍 𝑡 2 superscript subscript 𝑛 1 𝑚 italic-ϑ 𝑡 𝑡 𝑛 superscript 𝑛 1 2 𝑅 𝑡 {\displaystyle{\displaystyle{\displaystyle Z(t)=2\sum_{n=1}^{m}\frac{\cos\left% (\vartheta(t)-t\ln n\right)}{n^{1/2}}+R(t)}}} {\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


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