Formula:DLMF:25.16:E1

<< Dirichlet L-functions

formula in Mathematical Applications

Formula:DLMF:25.16:E2 >>

${\displaystyle{\displaystyle{\displaystyle\psi\left(x\right)=\sum_{m=1}^{\infty}\sum_{p^{m}\leq x}\ln p}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\psi}}}$ : Chebyshev ${\displaystyle{\displaystyle{\displaystyle\psi}}}$-function : http://dlmf.nist.gov/25.16#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2

Bibliography

Equation (1), Section 25.16 of DLMF.

URL links

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<< Dirichlet L-functions

formula in Mathematical Applications

Formula:DLMF:25.16:E2 >>