1. 0.1 ζ 1
0.1 ζ
0.1.1
ζ(s)
s ζ(s)
s=2
∞
1 1 1 π2
ζ(2) = (2) = 1 + + 2 + 2 + ··· = (1)
22 3 4 6
n=1
π2
π2
1735 Euler
1
1
∞
1 1 1
ζ(1) = (1) = 1 + + + + ···
2 3 4
n=1
∞
1 1 1
= (1 − p−1 )−1 = 1 ··· (2)
p
(1 − 2 ) (1 − 3 ) (1 − 5 )
1 1
=
∞ ∞
−s
ζ(s) = n = (1 − p−s )−1 (3)
n=1 p
1
(1)
Leibniz
1 1 1 1 1 π
1− + − + − + ··· =
3 5 7 9 11 4
1 1 1 1 π3
1− + 2 − 2 + 2 − ··· =
32 5 7 9 32
Dirichlet
1 1 1 1 1 1 1 √
1− − + + − − + · · · = √ log (1 + 2)
3 5 7 9 11 13 2
log

2. 0.1 ζ 2
Euler 2m (1)
B2m 2m
ζ(2m) = (−1)m+1 22m−1 π , m ∈ 1, 2, 3, . . . (4)
(2m)!
Bn
∞
t tn
t−1
= Bn (4)
e n!
n=1
2m × π 2m
2m B2m
B2m ζ(2m)
2m B2m B2m ζ(2m)
2 1 6 ζ(2) = 1 π 2 = 2×3 π 2
6
1
4 1 30 ζ(4) = 90 π = 2×312 ×5 π 4
1 4
1
6 1 42 ζ(6) = 33 ×5×7 π 6
1
8 1 30 ζ(8) = 2×33 ×52 ×7 π 8
1
10 5 66 ζ(10) = 35 ×5×7×11 π 10
691
12 691 2730 ζ(12) = 36 ×53 ×72 ×11×13 π 12
.
. .
. .
. .
.
. . . .
(1) 2
B2m
ζ(1 − 2m) = −
2m
(π )
2
Re(s) > 1
„ « „ «
s 1−s
π −2/3 Γ ζ(s) = π −(1−s)/2 Γ ζ(1 − s)
2 2
Z ∞
Γ(s) = e−x xs−1 dx
0
(4) 2m ≥ 2
B2m
ζ(1 − 2m) = −
2m

3. 0.1 ζ 3
虚数軸
●
非自明なゼロ点は実部1/2
オイラーの発見した ● の直線上にあるであろう
自明なゼロ点 リーマン予想
○ ○
-4 -2 -1 0 1/2 1 2
実軸
実部が１より大きい
領域にはゼロ点はない
●
●
Figure 1:
2m ζ(1 − 2m)
2 ζ(−1) = − 12 = − 221
1
×3
1 1
4 ζ(−3) = 120 = 23 ×3×5
6 ζ(−5) = − 252 = − 22 ×32 ×7
1 1
1 1
8 ζ(−7) = 240 = 24 ×3×5
10 ζ(−9) = − 132 = − 22 ×3×11
1 1
691 691
12 ζ(−11) = 32760 = 23 ×32 ×5×7×11
.
. .
.
. .
ζ(1 − 2m) = − B2m
2m 2m = n
ζ(−2) = 0, ζ(−4) = 0, . . .
ζ(0) = −1
2 s>1
• (3)
• f (s) = f (1 − s)
• 1/2
1/2 ( )

4. 0.1 ζ 4
0.1.2
(4)3
• 1735 ζ(2m + 1)
ζ(3) sin
4
• 1882 π
• 1979 ζ(3)
• 1884 -1885
• 2000 (ζ(s)
ζ(3), ζ(5), ζ(7), . . .
ζ(3), ζ(5), . . . , ζ(a) δ(a)
a→∞
log a
δ(a) ≥ (1 + O(1))
1 + log 2
• 2004 ζ(5), ζ(7), ζ(9), ζ(11)
5
3
W.Zudilin (4)
4
∞
X 1 (2πi)2k B2k
2 =− , k = 1, 2, 3, . . .
n=1
n2k (2k)!
(4)
4
5
K3
Picard-Fuchs Landau-Ginzburg
Sato-Tate Calabi-Yau
K3 Calabi-Yau Calabi-Yau
1

5. 0.1 ζ 5
Ramanujan 2006
∞
7π 3 1
ζ(3) = −2
180 n3 (e2πn − 1)
n=1
∞
19π 7 1
ζ(7) = −2
56700 n7 (e2πn − 1)
n=1
4m − 1
∞ 2m
1 1 B2j B4m−2j
ζ(4m − 1) = −2 2πn
(e − 1) − (2π)4m−1 (−1)j
n4m−1 2 (2j)!(4m − 2j)!
n=1 j=0
21
2003
6 sin Taylor
∞
(−1)m 2m+1 x3 x5
sin x = x =x− + ···
(2m + 1)! 3! 5!
m=0
sin x Z
∞
x2
sin x = x (1 − )
n2 π 2
n=1
x2 x2 x2
= x(1 − )(1 − 2 2 )(1 − 2 2 ) × · · ·
π2 2 π 3 π
∞
1 1
=x− 2 x3 + · · ·
π n2
n=1
x3 Taylor 1/6
1
x3 π2
ζ(2)
π2
ζ(2) =
6
6
9 20 I
I Euler (4)
2003

6. 0.1 ζ 6
n = 2, 4, 6, 8, . . .
ζ(5), ζ(7), ζ(9), ζ(11), . . .
7
0.1.3
1
• s ζ(s) =( )×π s
• s
• s ζ(s) = 0
• s ζ(s) =( )
Nr , Dr
1. 2/m 2m 4
2/m 2m 4
2. Dr p Dr
p−1 2m
1
ζ(−1) < 0, ζ(−3) > 0, ζ(−5) < 0, ζ(−7) > 0, ζ(−9) < 0, ζ(−11) > 0, . . .
s → −∞
2
2m + 1
7
Zagier ζk (s)
s=2 ζk (2) 3

7. 0.1 ζ 7
ζ(−11) 691
2m Nr Dr ζ(1 − 2m)
p =? p Dr
p − 1 2m
2 1 22 · 3 ζ(−1) = − 12 = − 221
1
×3
p=2
p=3
p=5
4 1 23 · 3 · 5 1 1
ζ(−3) = 120 = 23 ×3×5
p=2
p=3
p=5
p=7
6 1 22 · 32 · 7 ζ(−5) = − 252 = − 22 ×32 ×7
1 1
p=2
p=3
p=5
p=7
p = 11
8 1 24 · 3 · 5 1 1
ζ(−7) = 240 = 24 ×3×5
p=2
p=3
p=5
p=7
p = 11
10 1 22 · 3 · 11 ζ(−9) = − 132 = − 22 ×3×11
1 1
p=2
p=3
p=5
p=7
p = 11
p = 13
12 691 23 · 32 · 5 · 7 · 11 691 691
ζ(−11) = 32760 = 23 ×32 ×5×7×11
p=2
p=3
p=5
p=7
p = 11
p = 13
.
. .
. .
. .
.
. . . .

8. 0.1 ζ 8
0.1.4 691
Fermat 350
1994 Wiles
3 a
x=y=z=0
xa + y a = z a
19 Kummer
Q
• Dirichlet k ε ε−1 Ok
r = r1 +r2 −1
r1 r2
Ok ∼ Zr ⊕(
×
= )
•
Ik
Pk Cl(k) = Ik /Pk
19 1
√
• 1 k = Q( −m) 9
1967 Baker Stark
m = 1, 2, 3, 7, 11, 19, 43, 67, 163
1 m
Siegel
√
• k = Q( m) 1
Gauss

9. 0.1 ζ 9
Kummer Fermat
Kummer Q 1 n ξ
Q(ξ) (ξ n = 1) Q(e2πi/p ) p
p Q(e2πi/p )
1850
p h Q(e2πi/p ) h
p 8 p
B2m , m = 1, 2, . . . , (p − 3)/2 p
p xp + y p = z p x = y = z = 0
Kummer
• 1 2 ≤ 2m ≤ p − 3 2m
p
• 2 Q(e2πi/p ) p
p Fermat Kum-
mer 691 ζ(12)( ζ(−11))
Kummer x691 + y 691 = z 691
691
691
Kummer Herbrand Ribet
Kummer 2 Galois
∼
ω : σ ∈ Gal(Q(e2πi/p )/Q − → (Z/p)×
−
• 2’ Cl(Q(e2πi/p )) p x σ ∈
Gal(Q(e2πi/p )/Q
σ(x) = ω(σ)(1−r) x
2’ 1 1930 Herbrand 1
2’ Ribet 1976 Galois
Ribet
691 Ramanujan
∆ Ribet 1980
Frey Fermat
Fermat
8
h p 37, 59, 67, 101, . . .

10. 0.1 ζ 10
0.1.5
1 1
ζ(1) = 1 + + + ···
2 3
Riemann 1
Re s > 1
lim (s − 1)ζ(s) = 1
s→1+0
ζ(s) 1 s 1
9
Q F
Dirichlet
2r1 (2π)r2 hF
lim (s − 1)ζF (s) = RF
s→1 |DF | wF
ζF (s) hF
lim r1 +r2 −1
=− RF
s→0 s wF
9
9 20 I
1 1 1 1
log(2) = 1 − + − + − ...
2 3 4 5
Re s > 1
1 1 1 1
(1 − 21−s )ζ(s) = 1 − + s − s + s − ···
2s 3 4 5
lim (s − 1)ζ(s)
s→1+0
(s − 1)
= lim · (1 − 21−s )ζ(s)
s→1+0 1 − 21−s
1
= · log 2
log 2
=1
s=1
log
π log
π
log
log log
π

11. 0.1 ζ 11
r1 r2
hF F wF F 1
DF F RF F
α∈F
log |α(i)| (i = 1, . . . , r1 ),
l(i) α =
2 log |α(j)| (j = r1 + 1, . . . , r1 + r2 )
η1 , . . . , ηr
l(1) η1 l(1) η2 · · · l(1) ηr
l(2) η1 l(2) η2 · · · l(2) ηr
R[η1 , . . . , ηr ] =
··· ··· ··· ···
l(r) η1 l(r) η2 · · · l(r) ηr
(η1 , . . . , ηr )
• Beilinson
• Birch-Swinnerton-Dyer
Ramanujan Selberg Langlands
0.1.6
2005 Witten Langlands
6
Witten 6 4
4
S-
Schimmrigk q-
- Langlands
Witten Arthur 1989
Langlands
10
10