A
′
ζ(s) Re(s) = 1/2
M
II
25,1
II
25,1
ζ(s)
II
25,1
→ Λ
24
→ V
♮
→ j(τ) → T
2A
→ E
Γ
0
(2)
k
→ L(E
k
, s) = ζ(s) ·ζ(s−k+1) → ζ(s).
A
′
ζ(s)
II
25,1
II
25,1
(25, 1)
II
25,1
p
−1
Y
m>0, n∈Z
(1 − p
m
q
n
)
c(mn)
= j(σ) − j(τ),
c(n) j(τ) − 744 ρ = (0, 1, 2, . . . , 24; 70)
Λ
24
Λ
24
Θ
Λ
24
(τ) = 1 + 0 · q + 196560 q
2
+ 16773120 q
3
+ ···
0 q
1
Λ
24
= Λ
∗
24
Θ
Λ
24
(−1/τ) = τ
12
Θ
Λ
24
(τ).
Λ
24
∞ V
♮
V
♮
=
L
n≥−1
V
♮
n
dim(V
♮
n
) = c(n), j(τ) − 744 =
∞
X
n=−1
c(n) q
n
.
M V
♮
n
V
♮
1
196884 = 1 + 196883,
196883 M
T
2A
[g] M
T
g
(τ) =
∞
X
n=−1
Tr(g | V
♮
n
) q
n
T
2A
(τ) = q
−1
+ 4372 q + 96256 q
2
+ 1240002 q
3
+ ···
Tr(2A | V
♮
1
) = 1 + χ
196883
(2A) = 1 + 4371 = 4372 χ
196883
(2A) =
4371
Γ
0
(2)
T
2A
Γ
0
(2)
+
E
k
Γ
0
(2)
a
p
= 1 + p
k−1
p L
L(E
Γ
0
(2)
k
, s) = ζ(s) · ζ(s − k + 1) · ( 2).
Y
p
1
1 − (1 + p)p
−s
+ p
1−2s
=
Y
p
1
(1 − p
−s
)(1 − p
1−s
)
= ζ(s) · ζ(s − 1).
ζ(s)
ζ(s)
ζ(s) L
T
2A
j(τ) V
♮
Λ
24
II
25,1
II
25,1
→ Λ
24
Λ
24
→ V
♮
V
♮
→ j(τ) dim(V
♮
n
) = c(n)
j(τ) → T
2A
T
2A
→ E
Γ
0
(2)
k
T
2A
Γ
0
(2)
+
E
k
→ ζ(s) L
Λ
24
Θ
Λ
24
c(1) = 0
V
♮
dim(V
♮
1
) 196884 = 1 + 196883
T
2A
Tr(2A | V
♮
1
) 4372 = 1 + 4371
E
2
a
p
1 + p
ζ Re(ρ) = 1/2 ρ
g
σ
(t) = e
−t
2
/(2σ
2
)
W (g) = W
spec
+ W
arch
+ W
arith
σ W
spectral
W
archimedean
W
arithmetic
W (g
σ
)
−0.256
−1.341
−3.198
−6.276
−18.132
−50.165
W (g
σ
) > 0
ζ(s) Re(s) = 1/2
γ
1
= 14.1347 . . . , γ
2
= 21.0220 . . . , . . . , γ
30
= 101.3179 . . .
Re(ρ
n
) = 0.5
g ∈ S(R)
even
W (g) = W
spec
(g) + W
arch
(g) + W
arith
(g),
W
spec
(g) = 2 Re
Z
∞
0
t
−1/2
g(log t) dt
,
W
arch
(g) = −
Z
∞
0
K(t) g(t) dt, K(t) =
e
t/2
+ e
−t/2
e
t
− 1
−
2
t
,
W
arith
(g) = −
∞
X
n=1
Λ(n)
√
n
g(log n),
Λ(n) K(t) K(0) = 1/12
W (g ⋆ ˜g) ≥ 0 g ∈ S(R)
even
,
(g ⋆ ˜g)(x) =
R
R
g(x + t) g(t) dt
(0, ∞)
R
n
h : R → R
ˆ
h
h(0) > 0
h(r) ≤ 0 r ≥ r
0
ˆ
h(t) ≥ 0 t
h(r
0
) = 0 h
′
(r
0
) = 0
Φ : CE
24
→ S(R)
even
f ∈ CE
24
W
(Φf) ⋆
]
(Φf)
≥ 0.
Φ : CE
24
×N →
S(R)
even
f
24
∈ CE
24
Λ
24
g ∈ S(R)
even
N W ((Φ(f
24
, n)) ⋆
^
(Φ(f
24
, n))) ≥ 0
n ≥ N
W (g ⋆ ˜g) ≥ 0 g
A
′
Λ
24
Λ
24
= Λ
∗
24
Θ
Λ
24
Θ(−1/τ) = τ
12
Θ(τ) c(q
1
) = 0
∞ V
♮
j(τ) = j(−1/τ) V
♮
1
= 1 ⊕ 196883
T
2A
w
2
c(1) = 1 + 4371
E
k
L(f, s) ↔ L(f, k − s) a
p
= 1 + p
ζ(s) ξ(s) = ξ(1 − s) Re = 1/2
W (g ⋆ ˜g) =
X
ρ
|ˆg(ρ)|
2
| {z }
≥0
−
X
n≥1
Λ(n)
√
n
(g ⋆ ˜g)(log n)
| {z }
−
Z
∞
0
K(t) (g ⋆ ˜g)(t) dt
| {z }
.
A
′
D (Λ
24
, Θ
Λ
24
, )
(ζ(s), ξ(s) = ξ(1 − s), )
f g D W (g⋆˜g) ≥ 0
D g ∈ S(R)
even
f g
D(f, ·)
A
′
g f g
W (g ⋆ ˜g) ≥ 0 g
A
′
=⇒
A
′
=⇒
g ⋆ ˜g
E
8