Formula:DLMF:25.9:E1

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formula in Asymptotic Approximations

Formula:DLMF:25.9:E3 >>

${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{\sigma+\mathrm{i}t}=\sum_{1\leq n\leq x}\frac{1}{n^{s}}+\chi(s)\sum_{1\leq n\leq y}\frac{1}{n^{1-s}}+\BigO@{x^{-\sigma}}+\BigO@{y^{\sigma-1}t^{\frac{1}{2}-\sigma}}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle s=\sigma+\mathrm{i}t}}}$ &
${\displaystyle{\displaystyle{\displaystyle{\displaystyle\chi(s)=\pi^{s-\frac{1}{2}}\Gamma\left(\tfrac{1}{2}-\tfrac{1}{2}s\right)/\Gamma\left(\tfrac{1}{2}s\right)}}}}$ &
${\displaystyle{\displaystyle{\displaystyle t=2\pi xy}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle x\geq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle y\geq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle 0\leq\sigma\leq 1}}}$ &
formula valid as ${\displaystyle{\displaystyle{\displaystyle t\to\infty}}}$ with ${\displaystyle{\displaystyle{\displaystyle\sigma}}}$ fixed

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1

Bibliography

Equation (1), Section 25.9 of DLMF.

URL links

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<< Sums

formula in Asymptotic Approximations

Formula:DLMF:25.9:E3 >>