corrigé pdf

e2iπx e2iπx
5u u ∈U5
eiθk
θk=2iπ
5(x+k)k∈ {0,1,2,3,4}
Z=eiθ
iZ+ 1
Z−1=ieiθ + 1 eiθ −1= cotan θ 2
P
(X−i)5(i+ cotan(πx)) + (X+i)5(i−cotan(πx))
xsin πx 6= 0
i+ cotan πx =1
sin πx (cos πx +isin πx) = 1 sin πx eiπx
i−cotan πx =1
sin πx (−cos πx +isin πx) = −1 sin πx e−iπx
i−i
z P
z+i
z−i5
=e2iπx
h z →z+i
z−iC− {i}C− {1}x
e2iπx 6= 1 P h
e2iπx
z+i
z−i=Z z =iZ+1 Z−1= cotan θ 2Z=eiθ
θkP
cotan x+k
5π k ∈ {0,· · · ,4}
P
P5 4 0
P=eiπx
sin πx (X−i)5−e−iπx sin πx (X+i)5
=eiπx −e−iπx
sin πx X5+−eiπx −e−iπx sin πx 5iX4+· · · +eiπx +e−iπx sin πx (−i)
= 2iX5−5 cotan πx X4+· · · − cotan πx
X5−5 cotan πx X4+· · · − cotan πx =
4
Y
k=0 X−cotan x+k
5π
4 0
4
X
k=0
cotan x+k
5π= 5 cotan πx,
4
Y
k=0
cotan x+k
5π=−cotan πx
β(6) = 15
β(6) = 1 ∧6+2∧6+3∧6+4∧6+5∧6+6∧6 = 1 + 2 + 3 + 2 + 1 + 6 = 15
(σ∗µ)(12) = 12 µ(4 ×2) = 0 µ(2 ×3) = (−1)2= 1,· · ·
(σ∗µ)(12) = σ(1)µ(12)+σ(2)µ(6)+σ(3)µ(4)+σ(4)µ(3)+σ(6)µ(2)+σ(12)µ(1)
= 1 ×0 + (1 + 2) ×1 + (1 + 3) ×0 + (1 + 2 + 4) ×(−1) + (1 + 2 + 3 + 6) ×(−1)
+(1+2+3+4+6+12)×1 = 3 −7−12 + 28 = 12
e∗e=d n
(e∗e)(n) = X
d∈D(n)
e(d)
=1
e(n
d)
=1
=]D(n) = d(n)

I∗e=σ
(I∗e)(n) = X
d∈D(n)
I(d)
=d
e(n
d)
=1
=σ(n)
p φ(p) = p−1σ(p) = 1 + p d(p)=2
φ(p) + σ(p) = pd(p)
pmpkk0n
(µ∗e)(pm) = µ(1) + µ(p) + µ(p2) + · · ·
| {z }
=0
= 1 −1=0
δ=n
dC
(f∗g)(n) = X
δ∈D(n)
f(n
δ)g(δ) = X
δ∈D(n)
g(δ)f(n
δ) = X
d∈D(n)
g(d)f(n
d) = (g∗f)(n)
f e0(n)n= 1
e0∗f=f∗e0
∀n∈N∗,(e0∗f)(n)=(f∗e0)(n) = e0(1)f(n)⇒e0∗f=f∗e0=f
f g h n N∗
(d1, d2, d3)∈T(n)⇔(d1, d2d3)∈C(n)⇔d1∈D(n) (d2, d3)∈C(n
d1
)
⇔d1d2∈D(n
d3
)d3∈C(n)
(f∗(g∗h)) (n) = X
d1∈D(n)
f(d1)(g∗h)( n
d1
)
=X
d1∈D(n)
f(d1)
X
(d2,d3)∈C(n
d1)
g(d2)h(d3)
=X
(d1,d2,d3)∈T(n)
f(d1)g(d2)h(d3) = X
d3∈D(n)
X
(d1,d2)∈C(n
d3)
f(d1)g(d2)
h(d3)
=X
d3∈D(n)
(f∗g)( n
d3
)h(d3) = ((f∗g)∗h)(n)
∗
D(m)×D(n)D(mn)
P
d mn
P((a, b)) = d ab =d a ∈D(m)b∈D(n)a
d m a m ∧d b n ∧d
m∧d d mn m ∧d m
n m ∧d n
b n
m∧d d =ab
m∧d b)⇒m∧d a ⇒m∧d=a
n∧d=b
d P (m∧d, n ∧d)
m∧d

d m n ∧d
n
(m∧d)(n∧d) = d⇒P((m∧d, n ∧d)) = d
f g m n
(f∗g)(mn) = X
d∈D(mn)
f(d)g(mn
d) = X
(dm,dn)∈D(m)×D(n)
f(dmdn)g(m
dm
n
dn
)
=X
(dm,dn)∈D(m)×D(n)
f(dm)f(dn)g(m
dm
)g(n
dn
)dm∧dn= 1,m
dm
∧n
dn
= 1
=
X
dm∈D(m)
f(dm)g(m
dm)
X
dn∈D(n)
f(dn)g(n
dn)
= (f∗g)(m)(f∗g)(n)
I I(mn) = mn =I(m)I(n)m n
e0e0(mn)=0=I(m)I(n)m n
1
e e(mn) = 1 = e(m)e(n)m n
d P
σ=I∗e
I e
m n
f g nfng
f(nf)6= 0, g(ng)6= 0,∀k < nf:f(k)=0,∀k < ng: (k) = 0
k nfngk > nf
nfng
k< ngg(nfng k) = 0
k > ng
(f∗g)(nfng) (nf, ng)
(f∗g)(nfng) = f(nf)g(ng)6= 0
f∗g nfng
m < nfngk m
k≥nfm
k< ngg(m k) = 0 k < nff(k) = 0
(f∗g)(k)=0
N(f∗g) = N(f)N(g)
n pmp
m
n pkk≤m
(µ∗e)(n) =
m
X
k=0
µ(pk)e(pm−k) 0 1 = 1 + (−1) = 0
(µ∗e)(1) = µ(1)e(1) = 1
µ∗e=e0
f=g∗e⇒f∗µ= (g∗e)∗µ=g∗(e∗µ) = g∗(µ∗e) = g∗e0=g
g=f∗µ⇒g∗e= (f∗µ)∗e=f∗(µ∗e) = f∗e0=f
k∈F
δk ∈∆
(kδ)∧n= (kδ)∧(dδ) = δ(k∧d) = δ
k→δk
s∆δ=s∧n δ
s k s =δk
n=δd
s=δk
δ=s∧n
⇒d∧k= 1 ⇒k∈F

F∆
φ(d)φ
n∧x=a x 1n
a n
a n ∆
δ=a d =n
aφ(d) = φ(n a)
x1n n ∧x
d d φ(n
d)
n∧x=d
(n) = n=X
d∈D(n)
φ(n
d) = (e∗φ)(n) = (φ∗e)(n)
I=φ∗e
I=φ∗e⇒I∗µ= (φ∗e)∗µ=φ∗(e∗µ) = φ∗e0=φ
1n
n∧x
d
β(n) =
n
X
k=1
k∧n=X
d∈D(n)
φ(n
d)
|{z}
k k ∧n=d
d= (I∗φ)(n)
β=I∗φ⇒β∗e=I∗(φ∗e) = I∗I
σ=I∗e I =e∗φ
σ∗φ= (I∗e)∗φ) = I∗(e∗φ) = I∗I
I∗I
(I∗I)(n) = X
d∈D(n)
I(d)I(n
d) = X
d∈D(n)
dn
d=X
d∈D(n)
n=d(n)n
d(n)
φ+σ=φ∗σ
(φ∗σ)(n) = φ(1)
|{z}
=1
σ(n) + · · · +φ(n)σ(1)
|{z}
=1
· · · n
n1n φ σ
p
φ(p) + σ(p)=2p=pd(p)
φ(p) + σ(p)=2p=pd(p)
X2+X+ 1
X2−X+1
X2+ 2X+1
2
Ω ]0,π
m+1 [⊂Ω
θ∈Ω 2θ∈]0,2π[ sin 2θ > 0 2θ∈]0, π[θ∈]0,π
2[
θ∈]0,2
π[ 3θ∈]0,3π 2[ sin 3θ > 0θ∈]0,π 3[
θ∈]0,π
m+1 [
Dmu1u2C
Dm=X2(m+1) −2 Re(um+1
1)Xm+1 +|u1|2(m+1)
DmRe(um+1
1)
cos(m+ 1)φ < 0
Bm
Xm+1 −um+1
1= (X−u1)(Xm+u1Xm−1+· · · +um
1)
Xm+1 −um+1
2= (X−u2)(Xm+u2Xm−1+· · · +um
2)
Bm

Xk
Xm+u1Xm−1+u2
1Xm−2+· · · +um 1
um−k
1Xkum−k 2
k m u1=reiφ
bk=
k
X
α=0
um−α
1um−(k−α) 2=
k
X
α=0
r2m−kei(k−2α)φ
=r2m−keikφ
k
X
α=0
(e−2iφ)α=r2m−keikφ 1−e−2i(k+1)φ
1−e−2iφ =r2m−ksin(k+ 1)φ sin φ
b2m= 1, b2m−1=u1+u2, b2m−2=u2
1+u1u2+u2 2
b2m−k=uk
1+uk−1
1u2+· · · +u1uk−1 2+uk
2=uk+1
1−uk+1 2
u1−u2
=rksin(k+ 1)φ
sin φ
Bm
sin 2φ
sin φ,· · · ,sin(m+ 1)φ sin φ
φ u1Im u1>0 sin φ > 0
φ∈]0, π[Bm
φ∈]0,π
(m+ 1)[
Dmcos(m+ 1)φ < 0
π
2<(m+ 1)φ<π⇔π 2φ< m + 1 <π φ
φ
π
2≤φ C
B= 1 D=C
φ < π
21<π 2φ]π 2φ,π φ[π 2φ>1
2
m
C