la formule qui tue

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on désigne par:

$\varphi$ l’indicatrice d’Euler
$\psi$ la seconde fonction de Tchebychev
$\Lambda$ la fonction de Von Mangoldt

Montrez que :

$\displaystyle\frac{\displaystyle\sum_{d|e^{\psi(n)}}\ln(d)\varphi(d)}{\displaystyle\sum_{d|e^{\psi(n)}}\varphi(d)}=\psi(n)-\sum_{k=1}^{n}\frac{\Lambda(k)}{k}$

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