Exercise 60, § 1

a) Let $\mathbf{H}\subset \mathbf{C}$ be the open set of complex numbers $z$ such that $\mathscr{I}(z)>0$. The function

is holomorphic on H, and it satisfies $\Theta (z) = (z/i)^{-1/2}\Theta (-1/z)$ for $z\in H$, where $z\mapsto (z/i)^{1/2}$ is the unique holomorphic function on H which extends the function

$iy\mapsto \overline{y}$. (Use Exercise 16, b).)

b) Let $m$ and $n$ be strictly positive integers. One has

$k=0k=0$

(Apply the relation of the preceding question with $z=-2in/m+iy$ and let $y\rightarrow 0$.) c) For every integer $m\geqslant 1$ and every $n\in \mathbf{Z}$, put

For all strictly positive integers $m_1$ and $m_2$, one has

$((m-1)/2)^2\surd$

d) If $m$ is odd, one has $\tau (1, m) =i\overline{m}$. (Use a).)

e) If $p$ is an odd prime number, then $\tau (n, p) =\ell_p(n)\tau (1, p)$, where $\ell_p$ is the unique character of order 2 of $(\mathbf{Z}/p\mathbf{Z})^*$, extended to $\mathbf{Z}/p\mathbf{Z}$ by putting $\ell_p(0) = 0$.

f) Let $p\not=q$ be two odd prime numbers. One has

g) Deduce a new proof of the quadratic reciprocity law (cf. A, V, p. 156, exerc. 23): for all odd prime numbers $p\not=q$, one has