Représentation intégrale d`une série de Dirichlet

ζ
ζ, ζa, ζimp
ζλ, ζµ, ζα
ζβ
ζν
ζa
1
ez+1 ζa
ζaζimp
ζαζβ
ζα−3/2<R(s)<−1/2
NM
MM0
M
M N ζλ−3/2<R(s)<−1/2

P∞
n=1
a(n)
ns
s= 1
∞
X
n=1
a(n)
ns=ϕ(s)∞
X
n=1
b(n)
n1−s
(b(n))
∞
X
n=0
b(n)
z2+ (2n+ 1)2π2
(c(n))
∞
X
n=1
c(n)1
ez/n + 1
ζ
(a(n)) = (1,1,1,1, ...) (b(n)) = (1,1,1,1, ...) (c(n)) = (1,0,0,0, ...)
P∞
n=1
(−1)n−1
ns
(a(n)) = (1,−1,1,−1, ...) (b(n)) = (1,0,1,0, ...) (c(n)) = (1,0,0,0, ...)

ζ
ζ, ζa, ζimp
ζ
ζ(s) = ∞
X
n=1
1
nsR(s)>1
ζa
ζa(s) = ∞
X
n=1
(−1)n−1
nsR(s)>1
ζimp(s)
ζimp(s) = ∞
X
m=0
1
(2m+ 1)sR(s)>1
ζ
ζ(s) = 1
1−21−sζa(s)
ζ(s) = 1
1−2−sζimp(s)
ζλ, ζµ, ζα
λ λ(1) = 1 p λ(p) = −1
a b λ(ab) = λ(a)λ(b)λ
ζλ
ζλ(s) = ∞
X
n=1
λ(n)
nsR(s)>1
ζ(2s)ζ(s)
ζλ(s) = ζ(2s)
ζ(s)
ζλC
µ:N\ {0} → {−1,0,1}
µ(1) = 1
µ(n) = 0 p p2n
µ(n) = (−1)rn=p1···prr > 0pi
ζµ(s) = 1
ζ(s)=∞
X
n=1
µ(n)
nsR(s)>1

ζµζ−2k k
0≤R(s)≤1ζλ
0≤R(s)≤1s=−2k ζ(s)ζ(2s)
ζλ
ζλζµ
ζa
ζα
ζα(s) = ζa(2s)
ζa(s)
ζα(s) = (1 −21−2s)ζ(2s)
(1 −21−s)ζ(s)=1−21−2s 1−21−sζλ(s)
ζλ(s) = 1−21−s
1−21−2sζα(s)
1−21−s= 0 ⇐⇒ s= 1 + k2iπ
ln(2), k ∈Z
1−21−2s= 0 ⇐⇒ s=1
2+kiπ ln(2), k ∈Z
α
ζα(s) = ∞
X
n=1
α(n)
ns
(1 −21−s)ζα(s) = (1 −21−2s)ζλ(s)
∞
X
m=0
α(2m+ 1)
(2m+ 1)s=∞
X
m=0
λ(2m+ 1)
(2m+ 1)s
α(2m+ 1) = λ(2m+ 1)
2(2m+ 1)
∞
X
m=0
α(2(2m+ 1))
(2(2m+ 1))s−∞
X
m=0
2α(2m+ 1)
(2(2m+ 1))s=∞
X
m=0
λ(2(2m+ 1))
(2(2m+ 1))s
α(2(2m+ 1)) −2α(2m+ 1) = −λ(2m+ 1)
α(2(2m+ 1)) = λ(2m+ 1)