THÉORIE ANALYTIQUE DES NOMBRES EXERCICES 1

p6= 2
p≡1
p≡1( 4)
S={(x, y, z)∈N3:x2+ 4yz =p}
(x, y, z)7→ 
(x+ 2z, z, y −x−z)x<y−z
(2y−x, y, x −y+z)y−z < x < 2y
(x−2y, x −y+z, y)x > 2y
|S|
(x, y, z)7→ (x, z, y)
S
p≡1

≡5
p
x2+y2p x y p ≡1
Fpx2+y2
x
N p
N2+ 4 1
p
p6= 3 p=x2+ 3y2
p≡1
p6= 2 p=x2+ 2y2
p≡1
S P
N(S)
S
Πp∈S(1 −1
ps)−1=X
n∈N(S)
1
ns.
∞
X
k=1
zk
k.

lim
N→∞
N
X
n=1
1
N−log(N)
A⊂N
d(A) = lim
n→∞
# (A∩ {1,2,· · · , n})
n.
N
{n2:n∈N}
(3)?P
{a+kN :k∈N}.
N
a N a Z/NZ
χ:G(N) := (Z/NZ)∗→C
L(s, χ) =
∞
X
n=1
χ(n)
ns
(s)>1
χ G(N)
ψχ(s)≥1
(s)>1
X
p∈P
χ(p)
ps+ψχ(s)
log L(s, χ) (s)>1