Formula:DLMF:25.10:E1

<< Asymptotic Approximations

formula in Zeros

Formula:DLMF:25.10:E3 >>

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

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

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\mathrm{exp}}}}$ : exponential function : http://dlmf.nist.gov/4.2#E19
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{ph}}}}$ : phase : http://dlmf.nist.gov/1.9#E7
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\in}}}$ : element of : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9

Bibliography

Equation (1), Section 25.10 of DLMF.

URL links

We ask users to provide relevant URL links in this space.

<< Asymptotic Approximations

formula in Zeros

Formula:DLMF:25.10:E3 >>