### 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 ℜ(s) ≥ 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, 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 ≡ h (mod n).

### License

BSD License

### Topics

- Mathematics/Number theory
- Mathematics/Algebra

### Session Dirichlet_L

- Multiplicative_Characters
- Dirichlet_Characters
- Dirichlet_L_Functions
- Dirichlet_Theorem

### Depends on

- Bertrand’s postulate
- Dirichlet Series
- Finitely Generated Abelian Groups
- Landau Symbols
- The Hurwitz and Riemann ζ Functions

### Used by

- Three Squares Theorem
- Gauss Sums and the Pólya–Vinogradov Inequality

### Auto-related entries

- The Mason–Stothers Theorem

## Cite

×

Dirichlet_L-AFP

    @article{Dirichlet_L-AFP,
      author  = {Manuel Eberl},
      title   = {Dirichlet L-Functions and Dirichlet's Theorem},
      journal = {Archive of Formal Proofs},
      month   = {December},
      year    = {2017},
      note    = {\url{https://isa-afp.org/entries/Dirichlet_L.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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