E1 - Mathom

fkC∞R R∗
+
f0
k=−log kfkr f(r)
k= (−1)r(log k)rfk
α0fk
∀α∈R∗
+α≥α0⇒ |f(r)
k(α)| ≤ (log k)r
kα0+1 .
k
Pk≥1f(r)
k[α0; +∞[
α0r(Hr)S=Pk≥1fk
CrI= [α0; +∞[α
S(r)(α) = X
k≥1
f(r)
k(α).
(H0)Pk≥1fk(α0)
Pk≥1fkI fk
C1IPk≥1f0
kS C1I S0=Pk≥1f0 k
r S CrI S(r)=Pk≥1f(r)
k
Pk≥1f(r)
k(α0)f(r) kC1IPk≥1f(r+1) kI
Pk≥1f(r)
kIPk≥1f(r+1) kS(r)
S(r+1) =Pk≥1f(r+1)
k(Hr)
r S C∞I α0R∗
+S
C∞R∗
+
∀r∈NS(r)=X
k≥1
f(r)
k.
fk
fk
x y λ
∀k∈N∗x≤y⇒fk(x)≥fk(y)⇒
x≤y⇒X
k≥1
fk(x)≥X
k≥1
fk(y)

∀k∈N∗fk(λx + (1 −λ)y)≤λfk(x) + (1 −λ)fk(y)⇒
X
k≥1
fk(λx + (1 −λ)y)≤λX
k≥1
fk(x) + (1 −λ)X
k≥1
fk(y)
.
t7→ 1
tα+1 R∗
+k
∀t∈[k;k+ 1] 1
(k+ 1)α+1 ≤1 tα+1 ≤1 kα+1 .
t k k + 1
1
(k+ 1)α+1 ≤Zk+1
k
dt
tα+1 ≤1 kα+1 .
n N n
k=n N
N+1
X
k=n+1
1
kα+1 ≤ZN+1
n
dt
tα+1 .
N
ρα(n)≤Z∞
n
dt
tα+1
ρα(n)≤1
αnα.
n+ 1 N
ZN+1
n+1
dt
tα+1 ≤
N
X
k=n+1
1
kα+1
N
Z∞
n+1
dt
tα+1 ≤ρα(n)
1
α(n+ 1)α≤ρα(n).
1
α(n+ 1)α≤ρα(n)≤1 αnα.
n
0≤nα(Sα−σα(n)) = nαρα(n)≤1
α
Sα=σα(n)+o1
α
n= 1 σα(1) = 1 o(1) limα→∞ Sα=
1

Sα= 1 + ρα(1) ρα(1) −1
α
(1)
ρα(1) −1
α≤1−2−α α.
α t 7→ 2−tlog 2
h t R∗
+
h(t) = log t
tα+1 .
t[3; +∞[
h0(t) = 1−(α+ 1) log t
tα+2 .
t≥3 log t > log e= 1 h0[3; +∞[h
ρα(3) = S(α)−σα(3) α→ρα(3) ρ0
α(3)
S0(α) +
3
X
k=1
log k
kα+1
ρ0
α(3) = −
∞
X
k=4
h(k).
h
ρ0
α(3) ≥ −
∞
X
k=4 Zk
k−1
h(t)dt
≥ −Z∞
3
h(t)dt
≥ −−αlog t+ 1
α2tα∞
3
≥ −1 + αlog 3
α23α
α→ρα(3) −1
α(3α−1−αlog 3)/(α23α)
x x −1≥log x
x= 3αα→ρα(3) −1
αR∗
+
α→σα(3) ρα(3) −1
α=
(Sα−1
α) + σα(3) α
Sα−1
αα→σα(3)
n(1)
σα(n) + 1
αn−α−1≤Sα−1 α≤σα(n) + 1 α(n+ 1)−α−1

x α 7→ x−α−log x
n
X
k=1
1
k−log(n+ 1) ≤γ≤
n
X
k=1
1
k−log n .
0≤γ− n
X
k=1
1
k−log(n+ 1)!≤log(n+ 1) −log n= log(1 + 1 n)
γ= limn→∞ Pn
k=1 1
k−log(n+ 1)
Sαn ρα(n)
σα(n) (1) (n−α−(n+ 1)−α)/α
α= 0.5
21
√n−1 √n+ 1= 2 √n+ 1 −√n pn(n+ 1) =2 pn(n+ 1)(√n+√n+ 1) <1
n3/2
10−3n= 100
0.000993 S0.5
10−810−810−6
10−3S0.52.412874 ≤σ0.5(100) ≤2.413869
S0.510−32.611881
10−7n≥1014/3n≥46416
1/k3/2
S0.510−12
10−7
hαR∗
+
f
f(ta + (1 −t)b)≤tf(a) + (1 −t)f(b)t= 0 t= 1
(a, b)f a < b
Zb
a
f(t)dt ≤f(a) + f(b)
2.
f=hαa=k b =k+ 1
Zk+1
k
dt
tα+1 ≤1 21 kα+1 +1 (k+ 1)α+1 .
n N n k =n N
ZN+1
n
dt
tα+1 ≤1 2nα+1 +
N
X
k=n+1
1
kα+1 +1 2(N+ 1)α+1 .
N
1
αnα≤1 2nα+1 +ρα(n)

1
αnα−1 2nα+1 ≤ρα(n).
f
f(a)+(t−a)f0(a)≤f(t)t=a−h t =a+h a h h [a−h;a+h]
f
Za+h
a−h
f(t)dt ≥2hf(a).
f=hαa=k h = 1/2
Zk+1/2
k−1/2
dt
tα+1 ≥1 kα+1 .
n N n k =n+ 1
N
ZN+1/2
n+1/2
dt
tα+1 ≥
N
X
k=n+1
1
kα+1 .
N
ρα(n)≤Z∞
n+1/2
dt
tα+1 =1
αn+1
2α.
ρα(n)n
α= 0.5
1
αn+1
2α−1
αnα+1 2nα+1 =2√2
√2n+ 1 −2 √n+1
2√n3
= 2√2√2n−√2n+ 1
p2n(2n+ 1) +1
2√n3
=−2√2
p2n(2n+ 1)(√2n+√2n+ 1) +1
2√n3
=1
2√n
√2n+ 1(√2n+√2n+ 1) −4n
n√2n+ 1(√2n+√2n+ 1)
=1
2√np2n(2n+ 1) −(2n−1)
n√2n+ 1(√2n+√2n+ 1)
=1
2√n
2n(2n+ 1) −(2n−1)2
n√2n+ 1(√2n+√2n+ 1)(p2n(2n+1)+2n−1)
<1
2√n
6n−1
n√2n2√2n(4n−1)
<3
4n√n.3n=1
4n5/2
n n ≥(107/4)2/5n≥363 10−12
S0.50.9995 10−710−7
S0.5'2.612375298899 .