Dirichlet's Theorem

Def Dirichlet L-function The Dirichlet $L$-function is of the form: $L(s,\chi )=\sum _{n=1}^{\infty }\frac{\chi (n)}{n^s}$where $\chi$ is a Dirichlet character and $s$ a complex variable with real part greater than $1$.

Lemma Fix $(a,q)=1$. If $L(1,\chi )\ne 0$, then ${\lim }_{s\to 1^{+}}(s-1){\prod }_{\chi \mathrm{mod}q}L(s,\chi {)}^{\overline{\chi (a)}}\ne 0$Proof

Lemma Suppose $s>1$, then we have $\sum _{\chi \mathrm{mod}q}\overline{\chi (a)}\log L(s,\chi )=\phi (q){\sum }_{n\ge 1}{\sum }_{p^n\equiv a\mathrm{mod}q}\frac{1}{n{p}^{ns}}$Proof

Dirichlet's Theorem

For coprime positive integers $(a,q)=1$, there are infinitely many primes that are congruent to $a$ modulo $q$.

Proof