The Dirichlet L-functions are an extremely important tool in studying primes in arithmetic progressions – their zeros “control” distribution of primes in arithmetic progressions in the same manner as zeros of Riemann zeta function control the overall distribution of primes. The first and the most elementary result involving these zeros, which is the key result in most proofs of Dirichlet’s theorem, is that there is never a zero at point \( s=1\). This post will present a proof of this fact using results from algebraic number theory.
This post is based solely on the content of Marcus’s Number Fields. The prerequisities for it are basic results about ideals in number fields and a minute amount of complex analysis. No background in analytic number theory is necessary.