mathématiques 1

F
F(x) =
+∞
X
n=1
(−1)n−1
nx,
ζ]1,+∞[
ζ(x) =
+∞
X
n=1
1
nx.
F ζ
F
(gn)n≥1[0,1[
gn(t) =
n
X
k=0
(−t)k.
g(gn)
F(1) = Z1
0
g(t)t F (1)
X
n≥1
(−1)n−1
nx[2,+∞[F
+∞
F
x > 0 ]0,+∞[t7→ ln t
txln n nxn≥1
x
n≥1fn:x7→ (−1)n−1
nx
aX
n≥1
f0
n
[a, +∞[
FC1]0,+∞[

ζ
x > 1F(x)−ζ(x)x ζ(x)
F(x) = (1 −21−x)ζ(x).
ζ+∞
X
n≥1
anX
n≥1
bnX
n≥2
cncn=
n−1
X
k=1
akbn−k
xX
n≥2
cn(x)X
n≥1
(−1)n−1
nx
n x
FX
n≥2
cn(x)
x > 1
x > 0|cn(x)| ≥ 4x(n−1)
n2x
0< x ≤1
2X
n≥2
cn(x)
x= 1
x= 1
1
X(n−X)
cn(x)Hn−1
nHn= 1 + 1 2+· · · +1 n
Hn−1
nn≥2
X
n≥2
cn(x)
ζ1
ln 2 F0(1)
F x 7→ 1−21−x
a b ln 2 F0(1)
x1+
ζ(x) = a
x−1+b+o(1).
X
n≥1
vnvn[1,2]
vn(x) = 1
nx−Zn+1
n
t
tx.

n≥1x∈[1,2]
0≤vn(x)≤1
nx−1 (n+ 1)x.
x∈[1,2] X
n≥1
vn(x)γ=
+∞
X
n=1
vn(1)
x∈]1,2]
+∞
X
n=1
vn(x)ζ(x) 1 −x
X
n≥1
vn[1,2]
x1+
ζ(x) = 1
x−1+γ+o(1).
ln 2 γ
+∞
X
n=1
(−1)n−1ln n
n.
F(2k)
ζ(2k)k≥1
R[X] R
(Bn)R[X]
B0= 1,∀n∈N∗, B0
n=nBn−1Z1
0
Bn(t)t= 0.
(Bn)
bn=Bn(0) bnn
B1B2b1b2
n≥2Bn(1) −Bn(0)
n∈NBn(X) = (−1)nBn(1 −X)
k gkR R
gk(x) = B2kx
2πx∈[0,2π[gk2π.
(an(k))n≥0x
gk(x) = a0(k)
2+
+∞
X
n=1
an(k) cos(nx).
n≥1k≥1
an(k) = k
(nπ)2B2k−1(1) −B2k−1(0)−(2k)(2k−1) (2nπ)2an(k−1).

an(1) n≥1
n≥1k≥2
an(k) = (−1)k−1(2k)!
22k−1(nπ)2k.
k= 1
k≥1ζ(2k)b2k
bn
n∈N
Bn(X) =
n
X
k=0 n
kbn−kXk.
bnn