A <Dirichlet character> modulo $m$ is a homomorphism
$$
\chi:(\mathbb Z/m\mathbb Z)^\times\to\mathbb C^\times.
$$
Extend it periodically to $\mathbb Z$ by setting $\chi(n)=0$ when $(n,m)>1$. Its <Dirichlet L-function> is
$$
L(s,\chi)=\sum_{n\geq1}\frac{\chi(n)}{n^s}
=\prod_p(1-\chi(p)p^{-s})^{-1}
\qquad(\operatorname{Re}s>1).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 3 / a / Solution

{c}
{title2=$L(s,\chi)$}
{wiki}

For $\operatorname{Re}s>1$,
$$
L(s,\chi)=\sum_{n\geq1}\frac{\chi(n)}{n^s}
=\prod_p(1-\chi(p)p^{-s})^{-1}.
$$


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