Autour du théorème des nombres premiers

∗ †
π(x)x
π(x)∼
x
log x
1x π(x)
0,92129 x
log x+ox log x6π(x)61,10556 x log x+ox log x
π(x)∼x
log x
1
1
Pp
x
Pn6xP[x]
n=1 [x]x
∗
†

f g N?Rf ? g
f ? g(n) = X
ij=n
f(i)g(j) = X
i|n
f(i)gn
i
n∈N?G
1e1
e1(1) = 1 e1(n) = 0 n > 1
∈ G 1
µ ? µ(n)
n
µ
◦µ(1) = 1
◦µ(p1···pk) = (−1)kpi
◦µ(n) = 0 n
µ0? µ0=e1
Pd|nµ0(d) = e1(n)n n = 1
n > 1p n
a>1n n =n0pan0p d n
d=d0pbd0n006b6a
µ0(d0pb) = µ0(d0)µ0(pb)
X
d|n
µ0(d) = X
d0|n0
a
X
b=0
µ0(d0pb) = X
d0|n0
µ0(d0)·
a
X
b=0
µ0(pb).
a>1 1 −1+0+···+ 0 = 0
µ(nm) = µ(n)µ(m)n m
1
Pπ π(x) =
Pn6xP(n)
Λ(n) = log p n =pap a >1
0
n
x>1ψ(x) = Pn6xΛ(n)

P
ψ(x)
log p[1, x]
log x ψ(x)'π(x) log x
ΛP
?Λ = log µ ? Λ = −µlog
log
vn=unlog n
Punx>2
U(x) = X
n6x
un=u1+X
26n6x
unlog n1
log n=u1+V(x) log[x]+X
26n6x−1
V(n)1
log n−1 log(n+ 1) 
[x]x V (x) = Pn6xvnvn
V(x)
log[x]u1
x1
log n−1 log(n+1) ∼1
nlog2n
V(x) = O(x)
O(1
log2n)
X
26n6x
1
log2n=X
26n6√x
1
log2n+X
√x<n6x
1
log2n6√x log22+4x log2x=Ox log2x
U(x) = u1+V(x)
log[x]+Ox
log2x=V(x)
log x+Ox
log2x
V(x)
log[x]−V(x) log x
6|V(x)|1
log(x−1) −1 log x=O1
log2x.
un=P(n)
ψ(x)'π(x) log x
ψ(x) = X
p∈P
p6x
log plog x
log p6X
p∈P
p6x
log x=π(x) log(x)ψ(x)>V(x)
U(x)V(x)
V(x)
06V(x)6ψ(x)
ψ(x) = O(x)

?Λ = log n∈[1, x]
A(x) = X
d6xhx
diΛ(d) = X
n6x
log n=xlog x+O(x)
x
x
2
A(x)−2Ax
2=X
d6xhx
di−2hx 2diΛ(d) = xlog 2 + O(x) = O(x).
1d]x
2, x]
ψ(x)−ψ(x
2) = Px
2<d6xΛ(d) = O(x)xx
2
x
4
1
2n
ψ(x)
log x6π(x) = V(x) log x+Ox
log2x6ψ(x)
log x+Ox
log2x
π(x) = ψ(x)
log x+Ox
log2x.
π(x) = O(x
log x)
µ µTT∈N?
µT(n) = 
µ(n)n<T
−TPi<T
µ(i)
in=T
0n>T
µT(T)Pn
µT(n)
n= 0
gT=? µTGTGT(x) = Pn6xgT(n)x>1
1? µ =e1gT(n) = e1(n)n < T
GT(n) = 1 n<T
T∈N?α∈]0,1]
δ= 1 + X
j6T
log j
jµT(j) ; C= 1 + max
T6y< T
α|GT(y)|δ0= 2α+C
T.
δ0<1
lim sup
x→∞ 
π(x) log x
x−1
6|δ|+δ0
1−δ0.
T α
C δ δ0
µ µT

T α
(T, α)
T25 100 200 400 800
α0,005 0,010 0,010 0,010 0,002
|δ|+δ0
1−δ00,263 0,127 0,087 0,063 0,033
π(x) log x
x
ψ(x)
x
(T, α)C δ δ0
Λ? gT= Λ ? ? µT= log ?µT
X
n6x
Λ? gT(n) = X
n6x
log ?µT(n).
ψ(x)x
Pn6xf ?g(n)
f(i)g(j) (i, j)ij =x
j=
X
n6x
log ?µT(n) = X
j6x
µT(j)X
i6x
j
log i.
ix
jlog x j−
x
j+O(log x)
X
n6x
log ?µT(n) = (xlog x−x)X
j6x
µT(j)
j−xX
j6x
log j
jµT(j) + O(log x) = x−δx +O(log x)
x>T(δ−1)x
G
Λ(i)gT(j) (i, j)ij =x j 6T−1
gT(j) = e1(j) Λ(i)gT(j)ij 6x j 6T−1
Pi6xΛ(i) = ψ(x)G=ψ(x) + G0G0Λ(i)gT(j)
ij 6x j >T
G0=X
i6x
T
Λ(i)X
T6j6x
i
gT(j) = X
i6x
TGTx
i−GT(T−1)Λ(i) = X
i6x
TGTx
i−1Λ(i).
GT(x
i)x i<T αi > xα T
C
GT2T
y>1
GT(y) = X
j6Ty
jµT(j) = −X
j<T y
jµ(j) + Tny ToX
j<T
µ(j)
j