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\title{Dirichlet $L$-functions and Dirichlet's Theorem}
\author{Manuel Eberl}
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\begin{abstract}
This article provides a formalisation of Dirichlet characters and Dirichlet $L$-functions including proofs of their basic properties -- most notably their analyticity, their areas of convergence, and their non-vanishing for $\mathfrak{R}(s)\geq 1$. All of this is built in a very high-level style using Dirichlet series. The proof of the non-vanishing follows a very short and elegant proof by Newman~\cite{newman1998analytic}, which we attempt to reproduce faithfully in a similar level of abstraction in Isabelle.

This also leads to a relatively short proof of Dirichlet's Theorem, which states that, if $h$ and $n$ are coprime, there are infinitely many primes $p$ with $p \equiv h \pmod{n}$.
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