Formula:DLMF:25.9:E3

<< Formula:DLMF:25.9:E1

formula in Asymptotic Approximations

Formula:DLMF:25.10:E1 >>

${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{\tfrac{1}{2}+\mathrm{i}t}=\sum_{n=1}^{m}\frac{1}{n^{\frac{1}{2}+\mathrm{i}t}}+\chi\left(\tfrac{1}{2}+\mathrm{i}t\right)\sum_{n=1}^{m}\frac{1}{n^{\frac{1}{2}-\mathrm{i}t}}+\BigO@{t^{-1/4}}}}}$

Constraint(s)

formula valid as ${\displaystyle{\displaystyle{\displaystyle t\to\infty}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle O}}}$ : order not exceeding : http://dlmf.nist.gov/2.1#E3

Bibliography

Equation (3), Section 25.9 of DLMF.

URL links

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