Formula:DLMF:25.16:E2

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Formula:DLMF:25.16:E3 >>

${\displaystyle{\displaystyle{\displaystyle\psi\left(x\right)=x-\frac{\RiemannZeta^{\prime}@{0}}{\RiemannZeta@{0}}-\sum_{\rho}\frac{x^{\rho}}{\rho}+o\left(1\right)}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle x\to\infty}}}$ &
The sum is taken over the nontrivial zeros ${\displaystyle{\displaystyle{\displaystyle\rho}}}$ of ${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\psi}}}$ : Chebyshev ${\displaystyle{\displaystyle{\displaystyle\psi}}}$-function : http://dlmf.nist.gov/25.16#E1
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle o}}}$ : order less than : http://dlmf.nist.gov/2.1#E2

Bibliography

Equation (2), Section 25.16 of DLMF.

URL links

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