Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 137 / 2 / ii / Solution

Extend each Dirichlet character periodically to all integers by setting $\chi (n)=0$ when $\gcd (n,N)>1$. Its Dirichlet L-function is initially the absolutely convergent Dirichlet seriesThe group $(\mathbb{Z}/N\mathbb{Z}{)}^{\times }$ has $\phi (N)$ elements. Part (i) gives, for every integer $n$, including nonunits,For a nonunit both sides are zero, because the fixed class $a$ is a unit. Interchanging the finite character sum and the absolutely convergent Dirichlet series therefore provesThis is Orthogonality of Dirichlet characters applied to a residue-class indicator. Any meromorphic continuations of the two sides agree by uniqueness of analytic continuation.