Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 150 / 3 / b / Solution

For $\sigma >1$, both relevant Dirichlet series converge absolutely, so their product may be rearranged:The inner sum is one when $n=1$ and zero otherwise by Möbius inversion. HenceThis is also the reciprocal of the absolutely convergent Euler product.