Past exam of the analytic number theory and elliptic curves course of the University of Oxford / 2007 / A / 1 / i / Solution

The Dirichlet convolution is $(f_1\ast f_2)(n)={\sum }_{d\mid n}{f}_1(d)f_2(n/d)$. Fix $s$ with $\sigma =\mathrm{Re}s>1$. Absolute convergence of the two Dirichlet series givesThus the double series may be regrouped by $n=ab$, provingThe same estimate proves absolute convergence of the resulting Dirichlet series.