Λ
24
α = π/ log(196560) 196560 Λ
24
R
L
2
c
1
= 0 α
cos(αγ
n
) |cos | > 0.9
A
2
< D
4
< E
8
< Λ
24
α
χ
= π/ log(q)
Λ
24
Λ
24
E(z) = Ξ(z)·e
−iαz
⇐⇒
1
E
2
|E(¯z)| < |E(z)| (z) > 0
g 7→ gg
∗
L
2
σ
min
≈ 2.8 ×10
−16
Λ
24
√
2
Θ
Λ
24
(τ) = 1 + 196560q
2
+ 16773120q
3
+ ··· , q = e
2πiτ
,
c
1
= 0 q
1
√
2
L
k(L) α
L
= π/ log k(L)
dim α
L
π/α
L
A
2
D
4
E
8
Λ
24
α
c
1
= 0
ρ = 1/2 + iγ
n
ζ(s)
cos(αγ
n
)
|cos(αγ
n
)| > 0.9 |cos(αγ
n
)| < 0.1
|cos(αγ
1
)| = cos(0.258 · 14.135) ≈ 0.9996
|cos(αγ
n
)| = 0.6257
P
50
n=1
cos
2
(α
L
γ
n
) ≈ 25 = N/2 ⟨cos
2
⟩ = 1/2
A
2
c
1
= 0 α
cos(αγ
n
)
γ |cos(αγ)| ≈ 1
E(z) = Ξ(z) ·e
−iαz
,
Ξ(z) = ξ(1/2 + iz) α = π/ log(196560)
A(t) = Re[E(t)] B(t) = [E(t)] t ∈ R
⇐⇒
1
E(z) {z : (z) > 0}
2
2
|E(¯z)| ≤ |E(z)| (z) > 0
ϕ(t) = arg
Ξ(t)
− αt.
Ξ(t) t arg(Ξ(t)) π γ
n
ϕ
′
(t) = −α < 0 ϕ
2
R
24
f : R
24
→ R
f(0) > 0
f(x) ≤ 0 |x| ≥ 2
ˆ
f(t) ≥ 0 t
f(r
0
) = f
′
(r
0
) = 0 r
0
= 2
f
Λ
24
Φ f
Φ(f)(t) = f(e
t
) · e
t
· ψ(t),
ψ r
0
= 2
t = log 2 196560 · 2
−s
f(2) =
f
′
(2) = 0 t = log r t = log 2
log 2 c
2
· 2
−s
= 196560 · e
−s log 2
W (g g
∗
) = (g g
∗
)(0) log π
| {z }
−
X
n
Λ(n)
√
n
(g g
∗
)(log n)
| {z }
+
Z
K(t)(g g
∗
)(t) dt
| {z }
.
(g g
∗
)(0) ·
log π ≥ 0 (g g
∗
)(0) = ∥g∥
2
2
≥ 0
K(t) =
1
1−e
−2t
−
1
2t
t > 0
g 7→ g g
∗
L
2
Z
R
g(t)g(t − x
0
) dt = ⟨ (g), τ
x
0
( (g))⟩
L
2
,
S → L
2
τ
x
0
(Φ)
(Φ) S (R)
x
0
T T (x
0
) ̸= 0 T = 0 T = 0
T (x
0
) ̸= 0
{
ˆ
f
λ
: λ > 0}
f : R →
R
x
0
f R
f : R → R
S
λ>0
{x : f(x/λ) ̸= 0} R
f {y : f(y) ̸= 0}
x
0
∈ R ε > 0 y |y − x
0
| < ε f(y) ̸= 0 λ = 1
f(y/1) = f (y) ̸= 0 y
f(r) = p(r
2
)e
−πr
2
ˆ
f(ξ) = q(ξ
2
)e
−πξ
2
M → 50
σ
min
≈ 2.7963 × 10
−16
.
→
ˆ
f(t) ≥ 0
ˆ
f
→
ˆ
f(t) ≥ 0 t W (g g
∗
) ≥ 0
f
δ > 0 Π ≤ C
g = Π(f)
W (g g
∗
) ≥ (δ
2
− C
2
ε)∥
ˆ
f∥
2
1
≥ 0,
δ > C
√
ε
L(s, χ)
Re(s) = 1/2
α
χ
=
π
log q
, q = (χ).
α
χ
α q = 196560
q ≥ 3
α
q ∈ [3, 10) A
2
α > 1.36
q ∈ [10, 50) D
4
0.80 < α < 1.36
q ∈ [50, 7000) E
8
0.36 < α < 0.80
q ≥ 7000 Λ
24
α < 0.36
P
q≤50
cos
2
(α
q
·γ)
q ≤ 50 γ
α
χ
L(s, χ mod q)
α
χ
= π/ log q q
j
j(τ) = q
−1
+ 744 + 196884q + ···
196884 = 1 + 196883
196884 − 196560 = 324 = 18
2
= (24 − 6)
2
j
c = 24
Θ(−1/τ) = τ
12
Θ(τ)
ξ(s) = ξ(1 − s)
L(s, Θ
Λ
24
) =
Z
∞
0
Θ(it)t
s−1
dt ∼
X
c
n
n
−s
,
c
1
= 0
E
8