Logarithm In Terms of Von Mangoldt Function

Theorem
ln(n)=1Λ(d)=dnΛ(d)

where Λ is the von Mangoldt function and is the Dirichlet convolution.

Proof

Let n=p1α1pkαk as per the fundamental theorem of arithmetic. Then we have that

dnΛ(d)=dn{ln(p)d=pk for some k10otherwise}=i=1kj=1αiln(pi)=i=1kαiln(pi)=i=1kln(piαi)=ln(i=1kpiαi)=ln(n).