数学カフェ第１７回でtsujimotterが発表した資料です。 当日配布した「ゼータ関数」と「類数」の数表はこちらのURLに公開しています。ぜひ非正則素数で遊んでみてください。 http://tsujimotter.info/works/riemann_zeta_leftside/datasheet.pdf 第17回数学カフェ http://eventregist.com/e/xtO4W6ICsqUU tsujimotter のポートフォリオ http://tsujimotter.info/ tsujimotter のノートブック http://tsujimotter.hatenablog.com/

1. @tsujimotter
AQ(⇣691) ' Z/691Z Z/691Z
⇣( 11)

2. tsujimotter

4. •
•
•

5. tsujimotter

6. 1810 – 1893

8. ⇣(s) =
1X
n=1
1
ns (Re s > 1)

9. s
s
(s)

11. ⇣(1 s) = 21 s
⇡ s
sin
✓
⇡(1 s)
2
◆
(s)⇣(s)

12. ⇣(1 r) =
Br
r
r
Br r

13. 13
+ 23
+ 33
+ · · · + n3
12
+ 22
+ 32
+ · · · + n2
=
1
2
n2
+
1
2
n12
+ 22
+ 32
+ · · · + n2
=
1
3
n3
+
1
2
n2
+
1
6
n
=
1
4
n4
+
1
2
n3
+
1
4
n2
+ 0 · n
14
+ 24
+ 34
+ · · · + n4
=
1
5
n5
+
1
2
n4
+
1
3
n3
+ 0 · n2
-
1
30
n
n1

14. B0 = 1, B1 =
1
2
, B2 =
1
6
, B3
B0 = 1, B1 =
1
2
, B2 =
1
6
, B3 = 0
B0 = 1, B1 =
1
2
, B2 =
1
6
, B3 = 0, B4
0 = 1, B1 =
1
2
, B2 =
1
6
, B3 = 0, B4 = -
, B1 =
1
2
, B2 =
1
6
, B3 = 0, B4 = -
1
30
,

15. x
ex 1
=
1X
n=0
Bn
n!
xn
= 1
1
2
· x +
1
6 · 2!
· x2
+ 0 · x3 1
30 · 4!
· x4
+ 0 · x5
+ · · ·

16. ⇣( 1) =
1
12
=
1
22 · 3
⇣( 3) =
1
120
=
1
23 · 3 · 5
⇣( 5) =
1
252
=
1
22 · 32 · 7
⇣( 7) =
1
240
=
1
24 · 3 · 5
⇣( 9) =
1
132
=
1
22 · 3 · 11
⇣( 11) =
691
32760
=
691
23 · 32 · 5 · 7 · 13

17. ⇣( 11) =
691
32760
=
691
23 · 32 · 5 · 7 · 13
von-Staudt Clausen

18. von Staudt–Clausen
Dm =
Y
(p 1)|m
p
Bm

19. 1, 2, 3, 4, 6, 8, 12, 2424 !
⇣(1 24) =
B24
24
2, 3, 4, 5, 7, 9, 13, 25
D24 =
Y
(p 1)|24
p = 2 · 3 · 5 · 7 · 13
= 24 · 2 · 3 · 5 · 7 · 13⇣(1 24) =
B24
24

20. •
⇣( 23) =
236364091
65520
=
103 · 2294797
24 · 32 · 5 · 7 · 13

21. ⇣(1 r1) ⌘ ⇣(1 r2) (mod p)
r1 ⌘ r2 (mod p 1)
p r1, r2 r1 - p 1
A ⌘ B (mod p) A B
p

22. 1 321 68
⇣( 31) = 37 · 683 · 305065927/26
· 3 · 5 · 17
⇣( 67) = 37 · 101 · 123143 · 1822329343 · 5525473366510930028227481/23
· 3 · 5
r2 r1
r2 r1 ⌘ 0 (mod p 1)
p = 37
⇣(1 r2) ⇣(1 r1) ⌘ 0 (mod p)

23. p = 37
36363636
⇣( 31)⇣( 67)⇣( 103)⇣( 139)⇣( 175)
36 37

24. -1 -199 37

25. ⇣( 23) ⇣( 11) =
103 · 2294797
24 · 32 · 5 · 7 · 13
1
22 · 3
=
103 · 2294797 + 22
· 3 · 5 · 7 · 13
24 · 32 · 5 · 7 · 13
=
11 · 21488141
24 · 32 · 5 · 7 · 13
⇣( 23) ⇣( 11) =
103 · 2294797
24 · 32 · 5 · 7 · 13
1
22 · 3
=
103 · 2294797 + 22
· 3 · 5 · 7 · 13
24 · 32 · 5 · 7 · 13
=
11 · 21488141
24 · 32 · 5 · 7 · 13
⇣( 23) ⇣( 11) =
103 · 2294797
24 · 32 · 5 · 7 · 13
1
22 · 3
=
103 · 2294797 + 22
· 3 · 5 · 7 · 13
24 · 32 · 5 · 7 · 13
=
11 · 21488141
24 · 32 · 5 · 7 · 13
p = 11
10
⇣( 13)

26. mod p
mod pn
n = 1

27. p r1, r2 r1 - p 1
(1 1/p1 r1
)⇣(1 r1) ⌘ (1 1/p1 r2
)⇣(1 r2) (mod pn
)
r1 ⌘ r2 (mod (p 1)pn 1
)
n 1

28. p
⇣p(1 r) := (1 1/p1 r
)⇣(1 r)

29. p r1, r2 r1 - p 1
n 1
⇣p(1 r1) ⌘ ⇣p(1 r2) (mod pn
)
r1 ⌘ r2 (mod (p 1)pn 1
)

30. r1 ⌘ r2 (mod (p 1)pn 1
)
⇣p(1 r1) ⌘ ⇣p(1 r2) (mod pn
)
(p 1)pn 1
pn

31. y = f(x)
x
y

32. pn-1 pn

33. p
|x|p := p vp(x) p
x p
–1 –11 5 –1 –251 5
| 11 ( 1)|5 = | 10|5 =
1
5
=
1
5
| 251 ( 1)|5 = | 250|5
=
1
53

34. –1
–251
–11
–1 –11 5 –1 –251 5
| 11 ( 1)|5 = | 10|5 =
1
5
=
1
5
| 251 ( 1)|5 = | 250|5
=
1
53
=
1
53
=
1
5

35. p
r0
r1
r2
⇣p(1 r0)
⇣p(1 r1)
⇣p(1 r2)

36. •
• p

37. •
•
•

39. xn
+ yn
= zn
n 3
(xyz 6= 0)
(x, y, z)
FLT(n)

40. FLT(4)
FLT(3)
FLT(5)
FLT(7)
FLT(14)

41. p
FLT(p)

42. FLT(p)
FLT(3) FLT(5) FLT(7) FLT(11)
FLT(13) FLT(17) FLT(19) FLT(23) FLT(29)
FLT(31) FLT(37) FLT(41) FLT(43) FLT(47)
FLT(53) FLT(59) FLT(61) FLT(67) FLT(71)
FLT(73) FLT(79) FLT(83) FLT(89) FLT(97)

43. Q(⇣p)
Q(⇣p)
Z[⇣p]
Z[⇣p]
aq
q0 + q1⇣p + · · · + qp 2⇣p 2
p a0 + a1⇣p + · · · + ap 2⇣p 2
p

44. ⇣p

45. FLT
xp
+ yp
= zp
(x + y)(x + ⇣py)(x + ⇣2
p y) · · · (x + ⇣p 1
p y) = zp
Q(⇣p)
z = ✏pe1
1 · · · peg
g
= (✏pe1
1 · · · peg
g )p
x + ⇣k
p y = ✏0
↵p
(x ⇣k
p y), (x ⇣k0
p y)

46. ( )( ) = z2
= (P1P2)2
z = P1P2
= (P1P2)(P1P2)
= P2
1 P2
2

47. ( )( ) = z2
z = P1P2 = Q1Q2
= (P1P2)(Q1Q2)
= (P1Q1)(P2Q2)

48. p = 23
6 = 2 · 3 = ⇠1 · ⇠2
⇠1 = ⇣23 + ⇣4
23 + ⇣9
23 + ⇣16
23 + ⇣2
23 + ⇣13
23 + ⇣3
23 + ⇣18
23 + ⇣12
23 + ⇣8
23 + ⇣6
23
⇠2 = ⇣22
23 + ⇣19
23 + ⇣14
23 + ⇣7
23 + ⇣21
23 + ⇣10
23 + ⇣20
23 + ⇣5
23 + ⇣11
23 + ⇣15
23 + ⇣17
23

50. 6 = 2 · 3 = ⇠1 · ⇠2
A, B, C, D

51. Z
“ ”
3Z = (3)
3Z + 5Z = (3, 5)
“ ” “ ”

52. Z[⇣p]
2Z[⇣p] = (2)
3Z[⇣p] = (3)
⇠1Z[⇣p] = (⇠1)
⇠2Z[⇣p] = (⇠2) 3Z[⇣p] + ⇠2Z[⇣p] = (3, ⇠2)
3Z[⇣p] + ⇠1Z[⇣p] = (3, ⇠1)
2Z[⇣p] + ⇠1Z[⇣p] = (2, ⇠1)
2Z[⇣p] + ⇠2Z[⇣p] = (2, ⇠2)

54. (2)(3) = (6) (⇠1)(⇠2) = (⇠1⇠2)
(2, ⇠1)(3, ⇠2) = (2 · 3, 2⇠2, 3⇠1, ⇠1⇠2)
(3)(2, ⇠1) = (6, 3⇠1)

55. 積
(2, ⇠1)(2, ⇠2) = (22
, 2⇠1, 2⇠2, ⇠1⇠2)
= (22
, 2⇠1, 2⇠2, 6)
= (2)(1)
2, 3 = 1
= (2)(2, ⇠1, ⇠2, 3)
2
= (2)(1)

56. (2) = (2, ⇠1)(2, ⇠2)

57. (6) = (2)(3) = (⇠1)(⇠2)
(6) = (2, ⇠1) (2, ⇠2) (3, ⇠1) (3, ⇠2)

58. (↵)↵
Q(⇣p)
Q(⇣p) Q(⇣p)
A
A2
⇥A
A3
⇥A

60. •
•
•
•
JK PK
Cl(K) := JK PK
PK ⇢ JK
#Cl(K) K

61. Cl(Q(⇣p))
Cl(Q(⇣p))

62. xp
+ yp
= zp
(x + y)(x + ⇣py)(x + ⇣2
p y) · · · (x + ⇣p 1
p y) = zp
Q(⇣p)
(z) = pe1
1 · · · peg
g
= pe1
1 · · · peg
g
p
(x + y)(x + ⇣py)(x + ⇣2
p y) · · · (x + ⇣p 1
p y) = (z)p
(x + ⇣k
p y) = Ap
(x ⇣k
p y), (x ⇣k0
p y)

63. A = (↵)
(x + ⇣k
p y) = (↵p
) x + ⇣k
p y = ✏0
↵p
A = (↵)
(x + ⇣k
p y) = Ap

64. A
(x + ⇣k
p y) = Ap
#Cl (Q(⇣p))
pA = (↵)
p
A = (↵)
A = (↵)
p
p 1
1

65. #Cl(Q(⇣p)) p
#Cl(Q(⇣p)) p
FLT p
FLT p

66. #Cl (Q(⇣23)) = 3
#Cl(Q(⇣7)) = 1
#Cl(Q(⇣11)) = 1
#Cl(Q(⇣13)) = 1
#Cl(Q(⇣17)) = 1
#Cl(Q(⇣19)) = 1
#Cl(Q(⇣29)) = 8

67. #Cl(Q(⇣p)) = hp h+
p
p p

68. h7 = 1
h11 = 1
h13 = 1
h17 = 1
h19 = 1
h23 = 3
h29 = 8
h31 = 9
h37 = 37
37

69. 100
h37 = 371
h59 = 591
· 699
h67 = 671
· 12739
Remark

70. 200
h37 = 371
h59 = 591
· 699
h67 = 671
· 12739
h101 = 1011
· 35122815625
h103 = 1031
· 88049462555
h131 = 1311
· 217529616253985775
h149 = 1491
· 4616697044880367249149
h157 = 1572
· 2281404020463379154005

71. •
•
ex. ( ) hp p
•

72. •
•
•

73. p p

74. h37 37
⇣( 31) 37

75. h103 103
103⇣( 23)

76. p p
(1) p hp p
(2) p
⇣( 1), ⇣( 3), ⇣( 5), ⇣( 7), . . .

77. 691

79. ⇣( 1), ⇣( 3), . . . , ⇣(1 (p 3))
(2) p
(1) p hp p

80. 31
–1 –27
31

82. ⇣( 1), ⇣( 3), . . . , ⇣(1 (p 3))
(2) p
(1) p hp p

83. ⇣( 1), ⇣( 3), . . . , ⇣(1 (p 3))
(2) p

84. p-
p- p
Cl(Q(⇣p)) = AQ(⇣p) A0
Q(⇣p)
A!i
Q(⇣p) = {x 2 AQ(⇣p) | 8 2 , (x) = x!( )i
}
= Gal(Q(⇣p)/Q) !:
⇠
! (Z/pZ)⇥
AQ(⇣p) =
p 2M
i=0
A!i
Q(⇣p)

85. (1) (2)
(1)
(2)
p ⇣(1 r)
2 (x) = x!( )1 r
p xCl(Q(⇣p))
A!1 r
Q(⇣p) 6= {e}

86. p = 37
AQ(⇣37) ' Z/37Z
⇣( 31) =
7709321041217
16320
=
37 · 683 · 305065927
26 · 3 · 5 · 17
37
(x) = x!( ) 31
p
xk ⇠
! k. mod 37

87. p = 691
AQ(⇣691) ' Z/691Z Z/691Z
(x) = x!( ) 11
(y) = y!( ) 199
xk
· yl ⇠
! (k. mod 691, l. mod 691)
p
691⇣( 11), ⇣( 199)

88. 691

90. •
• p ó p
•

91. [1]
[2]
[3]
[1] [2] [3]

93. > zeta199.numerator
=>
49838404942833341476492863214039966210849588745720667496
80558226172636696215236875688658023022109991326014126976
13279391058654527145340515840099290478026350382802884371
712359337984274122861159800280019110197888555893671151
> zeta199.numerator % 691
=> 0
⇣( 199) ⌘ 0 (mod 691)

94. ↵A
↵(1) = (↵)
↵ 2 K⇥ A ⇢ OK

95. AB = (n)
A
B n
↵A ⇥
1
↵n
B =
↵
n↵
AB =
1
n
(n) = (1)

96. Sagemath
x = k.ideal(6)
x.factor()
(Fractional ideal (2, z^11 + z^10 + z^6 + z^5 + z^4 + z^2 + 1)) * (Fractional ideal (2, z^11 + z^9 + z^7
+ z^6 + z^5 + z + 1)) * (Fractional ideal (3, z^11 + z^10 + z^9 - z^8 - z^7 + z^5 + z^3 - 1)) *
(Fractional ideal (3, z^11 - z^8 - z^6 + z^4 + z^3 - z^2 - z - 1))
a = k.ideal(2, z^11 + z^10 + z^6 + z^5 + z^4 + z^2 + 1)
b = k.ideal(2, z^11 + z^9 + z^7 + z^6 + z^5 + z + 1)
c = k.ideal(3, z^11 + z^10 + z^9 - z^8 - z^7 + z^5 + z^3 - 1)
d = k.ideal(3, z^11 - z^8 - z^6 + z^4 + z^3 - z^2 - z – 1)
x = k.ideal(23)
x.factor()
z1 = z + z^4 + z^9 + z^16 + z^2 + z^13 + z^3 + z^18 + z^12 + z^8 + z^6
z2 = z^22 + z^19 + z^14 + z^7 + z^21 + z^10 + z^20 + z^5 + z^11 + z^15 + z^17
k.ideal(z1)
k.ideal(z1).reduce_equiv()
k.ideal(z2)
k.ideal(z2).reduce_equiv()
z1*z2