0022: Part 1, The 163 Dimensions

 
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Part 1: The 163 Dimensions of the Moonshine Functions

 

By Tito Piezas III

 

 

Keywords:  monstrous moonshine, moonshine functions, Monster group, prime-generating polynomials, McKay-Thompson series, pi formulas, 163.

 

Abstract:  An expository work on 1) Conway, Norton, and Atkin’s intriguing discovery that the moonshine functions span a linear space of dimension 163; and 2) the functions’ connection to Ramanujan-type pi formulas.

 

 

I. Introduction

II. The 172 McKay-Thompson Series and Linear Dependencies

III. More Pi Formulas?

IV. A Continued Fraction Using eπ√163

 

 

I.  Introduction

 

Consider the following prime-generating polynomials of form P(n) = an2-an+c,

 

P(n) = n2-n+41

P(n) = 2n2-2n+19

P(n) = 3n2-3n+23

P(n) = 4n2-4n+3

 

distinctly prime for n = {1 to c-1}, with the first (studied by Euler in 1772 in the form n2+n+41) for a remarkable 40 consecutive n.  And the following functions and their series,

 

1)  j(τ)  =  1/q + 744 + 196884q + 21493760q2 + 864299970q3 + …  (A007240)

 

2)  r2A(τ) = 1/q + 104 + 4372q + 96256q2 + 1240002q3 + …              (A101558)

 

3)  r3A(τ) = 1/q + 42 + 783q + 8672q2 + 65367q3 + …                        (A030197)

 

4)  r4A(τ) = 1/q + 24 + 276q + 2048q2 + 11202q3 + …                        (A097340)

 

where (throughout this article),

 

q = e2πiτ = exp(2πiτ) 

 

The first is the j-function j(τ), modular form of weight zero, but without the constant term, all these are simply the McKay-Thompson series T1A , T2A, T3A, and T4A of the Monster group.  Solving the polynomials for n at the point P(n) = 0, we get the complex roots,

 

n = (1+√-163)/2

n = (2+√-148)/4 = (1+√-37)/2

n = (3+√-267)/6

n = (4+√-32)/8 = (1+√-2)/2

 

where, consistent with the requirements for modular forms, we have chosen the root that is on the upper half of the complex plane. The first has fundamental discriminant -d = 163, the largest (in absolute value) with class number h(d) = 1, while the next two -d = {148, 267} = {4∙37,  3∙89} have class number h(d) = 2.  Let nτ, and plugging τ into q = exp(2πiτ), we get the very small real numbers,

 

q = -exp(-π √163)

q = -exp(-π/2 √148) = -exp(-π√37)

q = -exp(-π/3 √267)

q = -exp(-π/4 √32) = -exp(-π√2)

 

which, when substituted into the appropriate series above, yields the integers,

 

1)  j((1+√-163)/2) = -6403203

 

2)  r2A((1+√-37)/2) = -141122

 

3)  r3A((3+√-267)/6) = -3003

 

4)  r4A((1+√-2)/2) = -27

 

The first integer appears in the Chudnovsky brothers’ pi formula,

 

 

where one can see the 163, as well as in the famous, 

 

eπ√1636403203 + 743.9999999999992…

 

The second integer appears in a pi formula found by Ramanujan in 1912 (before the Monster group was even discovered),

 

 

where 4(37) also appears, as well as in the almost-integer,

 

eπ/2√148 = eπ√37141122 + 103.99997…

 

This (and similar formulas) inspired the one found by Chudnovsky brothers.  The third appears in another kind of pi formula,

 

 

where, one can see |d| = 267, and in the almost-integer,

 

eπ/3√267 = eπ√(89/3)3003 + 41.99997…

 

And finally, -27 appears in a fourth kind,

 

 

Using his own approach, Ramanujan found examples for all four types.  (See Pi Formulas and the Monster Group for the complete list of 36 known formulas where the denominator C is an integer.)  We instead used four functions based on T1A, T2A, T3A, T4A, but there are in fact many more of them.  And here’s a the astonishing fact first noticed by Conway and Norton:  the functions [1], [2], [3], [4], and others, call these the moonshine functions, span a linear space of dimension 163!  Interesting “coincidence” that it had to be that number, isn’t it?


 

II. The 172 McKay-Thompson Series and Linear Dependencies

 

To recall, the Monster group has 194 conjugacy classes so its character table has 194 rows and 194 columns.  Each column represents one conjugacy class and linear combinations of the entries gives its McKay-Thompson series which yields, to quote John McKay or Mark Ronan, a “moonshine function” (hence the title of this article).  Some are the same so is reduced to 172 series, with the two distinct conjugacy classes A & B of order 27 still included as an exceptional case.  (Author note: My thanks to Michael Somos for clarifying certain points.)

 

 

1

T1A

 

44

T12J

 

87

T25A

 

130

T46A

2

T2A

 

45

T13A

 

88

T26A

 

131

T46C

3

T2B

 

46

T13B

 

89

T26B

 

132

T47A

4

T3A

 

47

T14A

 

90

T27A

 

133

T48A

5

T3B

 

48

T14B

 

91

T27B

 

134

T50A

6

T3C

 

49

T14C

 

92

T28A

 

135

T51A

7

T4A

 

50

T15A

 

93

T28B

 

136

T52A

8

T4B

 

51

T15B

 

94

T28C

 

137

T52B

9

T4C

 

52

T15C

 

95

T28D

 

138

T54A

10

T4D

 

53

T15D

 

96

T29A

 

139

T55A

11

T5A

 

54

T16A

 

97

T30A

 

140

T56A

12

T5B

 

55

T16B

 

98

T30B

 

141

T56B

13

T6A

 

56

T16C

 

99

T30C

 

142

T57A

14

T6B

 

57

T17A

 

100

T30D

 

143

T59A

15

T6C

 

58

T18A

 

101

T30E

 

144

T60A

16

T6D

 

59

T18B

 

102

T30F

 

145

T60B

17

T6E

 

60

T18C

 

103

T30G

 

146

T60C

18

T6F

 

61

T18D

 

104

T31A

 

147

T60D

19

T7A

 

62

T18E

 

105

T32A

 

148

T60E

20

T7B

 

63

T19A

 

106

T32B

 

149

T60F

21

T8A

 

64

T20A

 

107

T33A

 

150

T62A

22

T8B

 

65

T20B

 

108

T33B

 

151

T66A

23

T8C

 

66

T20C

 

109

T34A

 

152

T66B

24

T8D

 

67

T20D

 

110

T35A

 

153

T68A

25

T8E

 

68

T20E

 

111

T35B

 

154

T69A

26

T8F

 

69

T20F

 

112

T36A

 

155

T70A

27

T9A

 

70

T21A

 

113

T36B

 

156

T70B

28

T9B

 

71

T21B

 

114

T36C

 

157

T71A

29

T10A

 

72

T21C

 

115

T36D

 

158

T78A

30

T10B

 

73

T21D

 

116

T38A

 

159

T78B

31

T10C

 

74

T22A

 

117

T39A

 

160

T84A

32

T10D

 

75

T22B

 

118

T39B

 

161

T84B

33

T10E

 

76

T23A

 

119

T39C

 

162

T84C

34

T11A

 

77

T24A

 

120

T40A

 

163

T87A

35

T12A

 

78

T24B

 

121

T40B

 

164

T88A

36

T12B

 

79

T24C

 

122

T40C

 

165

T92A

37

T12C

 

80

T24D

 

123

T41A

 

166

T93A

38

T12D

 

81

T24E

 

124

T42A

 

167

T94A

39

T12E

 

82

T24F

 

125

T42B

 

168

T95A

40

T12F

 

83

T24G

 

126

T42C

 

169

T104A

41

T12G

 

84

T24H

 

127

T42D

 

170

T105A

42

T12H

 

85

T24I

 

128

T44A

 

171

T110A

43

T12I

 

86

T24J

 

129

T45A

 

172

T119A

 

Table 1. The 172 McKay-Thompson series of Monstrous Moonshine

 

The first dozen coefficients of each of the 172 series are given in Table 4, Values of Head Characters, (p. 334) of John Conway and Simon Norton’s 1979 paper, Monstrous Moonshine.  Conveniently, an index of the McKay-Thompson series is also in the Online Encyclopedia of Integer Sequences.  (The moonshine functions, in accordance with Atlas notation, have the letter corresponding to the conjugacy class in upper-case, while the non-monstrous ones have it in lower-case.  By accident, T24B is missing in the index, but is still in the OEIS.) 

 

Also, David Madore has calculated the first 3500 coefficients of all the 172 moonshine McKay-Thompson series, while Michael Somos has an extensive database about the relationships between the coefficients and more.

 

But there are linear dependencies between the 172 series,

 

#

Linear dependency

 

1

 

T27A = T27B

 

2

 

T6A + 2T6E = T6B + T6C + T6D

 

3

 

T10A + 2T10E = T10B + T10C + T10D

 

4

 

T12A + 2T12I = T12B + T12E + T12H

 

5

 

T18B + 2T18D = T18A + T18C + T18E

 

6

 

T30B + 2T30G = T30A + T30C + T30F

 

7

 

2T8E = T4C + T4D

 

8

 

2T16B = T8D + T8E

 

9

 

T12C + T12E - T12F  - T12G - 2T12I = 2(T24B - T24C - T24I)

 

Table 2. The 9 Linear Dependencies

 

 

Taking into account these nine linear dependencies, then the dimension can be reduced to 172 – 9 = 163.  To bring this insight to life, in his blog, Lieven le Bruyn quotes p. 227 of Mark Ronan’s book Symmetry and the Monster,

 

Conway recalls that, ‘As we went down into the 160s, I said ‘...let’s guess what number we will reach.’ They guessed it would be 163 – which has a very special property in number theory – and it was! There is no explanation for this. We don’t know whether it is merely a coincidence, or something more....”

 

The first six were found by Conway and Norton, while the last three were discovered by Oliver Atkin who also showed that there are no more.  These nine are found in Conway and Norton’s paper, but in rather technical form.  For example, dependency # 7 is given as,

 

T4- + T4|2 = 2T8-

 

so I had to translate it into English, so to speak, i.e. convert it into the notation used in the OEIS (as well as in Madore’s and Somos’ databases).  We then find from the OEIS,

 

T4C(q) = 1/q + 20q - 62q3 + 216q5 - 641q7  + …      (A007248) 

 

T4D(q) = 1/q - 12q + 66q3 - 232q5 + 639q7  + …      (A007249)

 

T8E(q) = 1/q  +  4q  +  2q3  -  8q5  - q7   + …            (A029841)

 

and we can easily see from the first few coefficients that indeed,

 

T4C + T4D = 2T8E

 

though it is proven by Atkin that the equality holds for all terms.  Thus, just as one can consider the series T27B as redundant being equal to T27A, and since there are nine linear dependencies, then only 163 series as “moonshine functions” are really necessary to linearly express all possible values of the 172 series.  Or, as was mentioned, these functions span a linear space of dimension 163. 

 

In an email, McKay calls this as a “delicious coincidence”, while Norton opines this may be merely a coincidence.  However, in Conway and Norton’s paper, in the last paragraph of Section 9 which deals with moonshine for other groups, it was asked if there is “…a period three automorphism for the case Γ0(67)+”.  (The discriminant d = -67 is, of course, the younger brother of d = -163 in the family of Heegner numbers.)


 

III. More Pi Formulas?

 

In addition to the very interesting fact that the moonshine functions span a linear space of dimension 163, one question we can ask is, since there are so many of them, it is possible to use other McKay-Thompson series TpA in Ramanujan-type pi formulas?  Ramanujan’s four kinds of pi formulas are of the form,

 

 

where hp is one of various products of three Pochhammer symbols (their factorial equivalents are given in this article) and C is either j(τ), r2A(τ), r3A(τ), r4A(τ) or equivalently, the McKay-Thompson series T1A, T2A, T3A, T4A with an appropriate constant term.  It is tempting to speculate that, perhaps using different Pochhammer symbols and an appropriate constant term, one can use, say, T7A as well. 

 

In fact, J. Guillera has extended Ramanujan’s results and found pi formulas of form,

 

 

where gp is one of various products of  five Pochhammer symbols, but apparently the denominators are not values of any McKay-Thompson series.  See “A Compilation of Ramanujan-Type Formulas for 1/πm” for m > 1 for a list.

 

 

IV. A Continued Fraction Using eπ√163

 

First, given the curious prime-generating polynomial,

 

P(n) = 4n2+163

 

which is prime for n = {0 to 19}.  Solve for P(n) = 0, hence n = (1/2)√-163, plug this into r4A(n) given in the Introduction and we find that it is an algebraic number,

 

r4A((1/2)√-163) = x24

  

where x ≈ 5.31863…  is the real root of the simple cubic equation x3-6x2+4x-2 = 0Second, given the Ramanujan constant R,

 

R = eπ√163 ≈ 6403203 + 743.9999999999992…

 

and we see that R and x have the beautiful relationships,

 

 

and,

 

 
where q = -1/R = -1/eπ√163.  The root x has the exact value, let τ = (1+√-163)/2,

 

 

where η(τ) is the Dedekind eta function which partly explains the two relationships between R and x.  While this continued fraction is not unique to d = 163 (the general form is by E. Heine, 1846), I thought of including it in this article since R is a well-known constant. The significance of the integer 24 will be further elaborated in other articles.  Furthermore, it is also the case that,

 

eπ√43 ≈ 123(92-1)3 + 743.999…

eπ√67 ≈ 123(212-1)3 + 743.99999…

eπ√163 ≈ 123(2312-1)3 + 743.999999999999…

 

The reason for the squares within the cubes is a certain Eisenstein series.  But that’s another story ...
 

 

 

-- End --

 
 

© July 2011, Tito Piezas III

You can email author at tpiezas@gmail.com.

 

  

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