Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 150 / 1 / a

Let $1(n)=1$. The basic Von Mangoldt divisor identity isbecause if $n={\prod }_p{p}^{v_p(n)}$, the right-hand side is ${\sum }_p{v}_p(n)\log p=\log n$. In terms of Dirichlet convolution, this says $\log =1\ast \Lambda$. Since the Möbius function is the convolution inverse of $1$, convolving with $\mu$ gives $\Lambda =\mu \ast \log$. Consequently

Solved by gpt-5.6-sol high.