Exercise 17, § 1

Soient $q\geqslant 1$ an integer, and let $a\geqslant 1$ be an integer prime to $q$. Let $\widetilde{\chi}$ be a character of the group $(\mathbf{Z}/q\mathbf{Z})^*$. We denote by $\chi$ the mapping of $\mathbf{Z}$ into $\mathbf{C}$ such that $\chi (n) = 0$ if $n$ is not prime to $q$ and $\chi (n) =\widetilde{\chi}(n$ (mod$. q$)) otherwise (« Dirichlet characters »).

a) Suppose that $\widetilde{\chi}$ is nontrivial. The function defined by

is holomorphic on the open set U of $\mathbf{C}$ formed by the complex numbers such that $\mathscr{R}(s)>1$, and there exists a unique holomorphic function on $\mathscr{R}(s)>0$ which coincides with it on U. One has

for $s\in U$, where the product is taken over the prime numbers and converges absolutely and uniformly on the compact parts of U (« Dirichlet L Function »).

b) Suppose that $\widetilde{\chi}^2$ is not trivial. One has $L(1, \chi )\not= 0$. (Adapt the method of question j) of the preceding exercise.)

c) Suppose that $\widetilde{\chi}$ is of order 2. For every integer $n\geqslant 1$, denote

where the sum is taken over the divisors $d\geqslant 1$ of $n$. There exists a real number $c >0$ such that, for every $x\geqslant 2$, one has

$\sum r_{\chi}\surd (\frac{n}{n})\geqslant c$ log($x$)

$n\leqslant x$

(note that the mapping $r_{\chi}$ is $\geqslant 0$ and multiplicative, that is to say that $r_{\chi}(nm) =r_{\chi}(n)r_{\chi}(m)$ if $n$ and $m$ are prime to each other, and note that $r_{\chi}(n)\geqslant$ 1 if $n$ is a square of an integer.)

d) We suppose still that $\widetilde{\chi}$ is of order 2. One has

$n\leqslant x$

when $x\rightarrow +\infty$. Deduce that $L(1, \chi )\not= 0$. e) When $\sigma >1$ tends to 1, one has

$p\equiv a$ mod$. qp$

where $\varphi (q)$ is the Euler indicator of $q$ (A, V, p. 76).

f) There exists an infinity of prime numbers congruent to $a$ modulo $q$ (« theorem of the arithmetic progression of Dirichlet »).