Partie I. Une formule d`Euler. Partie II. Constante d`Euler.

x δ 1
xbxcx{x}
x=bxc+{x} {x} ∈ [0,1[
xE(x)
xP(x)
P(x)π(x)
P(x)N(x)
m
P(x)
mNm(x)
δ≥1
Zx=X
n∈E(x)
1
nδSm(x) = X
n∈Nm(x)
1
nδ
δ > 1x≥2
E(x)N(x)Nm(x)
x m E(x)⊂ Nm(x)
x m y > 1Nm(x)⊂ E(y)
Sm(x)ZxZy
Y
p∈P(x) m
X
k=0
1
pkδ !=Sm(x)
(Zi)i∈N∗
j
δ−1
(j+ 1)δ≤1 jδ−1−1 (j+ 1)δ−1≤δ−1 jδ
1
δ−1−1 (δ−1)(i+ 1)δ−1≤Zi≤1 δ−1+ 1 −1 (δ−1)iδ−1
(Zi)i∈N∗ζ(δ)
1
δ−1≤ζ(δ)≤1 δ−1+ 1
x(Sm(x))m∈N∗S(x)
S(x)≤ζ(δ)
p
m
X
k=0
1
pkδ !m∈N∗
Zx≤Y
p∈P(x)
1
1−1
pδ≤ζ(δ)
lim
x→∞ Y
p∈P(x)
1
1−1
pδ
=ζ(δ)
δ= 1
Zx=X
n∈E(x)
1
n

n un=Zn−ln n
i
1
i+ 1 ≤ln(i+ 1) −ln(i)≤1 i
(un)n∈N∗
∀n∈N∗:1
n≤un
(un)n∈N∗γ vn=un−γ
n
∀x∈[0,1[ : x+ ln(1 −x) + x2
2(1 −x)≥0
∀n∈N∗:1
n+ 1 + ln 1−1 n+ 1+1 2n(n+ 1) ≥0
n wn=vn−1
2n(wn)n∈N∗
∀n∈N∗: 0 ≤Zn−ln n−γ≤1
2n
∀x∈[0,1[ : x+ ln(1 −x) + x2
2(1 + x)≤0
∀n∈N∗:1
n+ 1 + ln 1−1 n+ 1+1 2(n+ 1)(n+ 2) ≤0
vn−1
2(n+1) n∈N∗
Zn−ln n−γ∼1
2n
u
v
m
uv =x

δ= 1 n1τ(n)
n D(x)
x+∞
D(x)
x=1 xX
n∈E(x)
τ(n) = ln x+ 2γ−1 + O(1
√x)
f g h ]1,+∞[
+∞:f(x) = g(x) + O(h(x))
Hxτ(x)D(x)
2X
m∈E(√x)bx
mc=D(x) + b√xc2
xZ√x−√x≤X
m∈E(√x)bx
mc ≤ xZ√x
∀n∈N\ {0,1}, θ(n)≤nln 4 θ(n) = X
p∈P(n)
ln p
n= 2
n≥4n−1
n
n n = 2m+ 1 m∈N
(1 + 1)2m+1
2m+ 1
m≤4m
p m + 1 < p ≤2m+ 1
p2m+1
m
θ(2m+ 1) −θ(m+ 1) ≤ln 2m+ 1
m
n≥2