Formula:DLMF:25.10:E3

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

Name

Riemann-Siegel formula

Note(s)

${\displaystyle{\displaystyle{\displaystyle R(t)=\BigO@{t^{-1/4}}}}}$ as ${\displaystyle{\displaystyle{\displaystyle t\to\infty}}}$ & ${\displaystyle{\displaystyle{\displaystyle{\displaystyle R(t)=(-1)^{m-1}\left(\frac{2\pi}{t}\right)^{1/4}\frac{\cos\left(t-(2m+1)\sqrt{2\pi t}-\frac{1}{8}\pi\right)}{\cos\left(\sqrt{2\pi t}\right)}+\BigO@{t^{-3/4}}}}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle O}}}$ : order not exceeding : http://dlmf.nist.gov/2.1#E3
${\displaystyle{\displaystyle{\displaystyle(-1)}}}$ : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4

Bibliography

Equation (3), Section 25.10 of DLMF.

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