Logarithm In Terms of Von Mangoldt Function Theorem ln(n)=1∗Λ(d)=∑d∣nΛ(d) where Λ is the von Mangoldt function and ∗ is the Dirichlet convolution. ProofLet n=p1α1…pkαk as per the fundamental theorem of arithmetic. Then we have that∑d∣nΛ(d)=∑d∣n{ln(p)d=pk for some k≥10otherwise}=∑i=1k∑j=1αiln(pi)=∑i=1kαiln(pi)=∑i=1kln(piαi)=ln(∏i=1kpiαi)=ln(n).