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 e^π√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) r_2A(τ) = 1/q + 104 + 4372q + 96256q² + 1240002q³ + … (A101558)

3) r_3A(τ) = 1/q + 42 + 783q + 8672q² + 65367q³ + … (A030197)

4) r_4A(τ) = 1/q + 24 + 276q + 2048q² + 11202q³ + … (A097340)

where (throughout this article),

q = e^2πiτ = exp(2πiτ)

The first is the j-function j(τ), a modular form of weight zero, but without the constant term, all these are simply the McKay-Thompson series T_1A , T_2A, T_3A, and T_4A 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 n = τ, and plugging those into the appropriate series above, yields the integers,

1) j((1+√-163)/2) = -640320³

2) r_2A((1+√-37)/2) = -14112²

3) r_3A((3+√-267)/6) = -300³

4) r_4A((1+√-2)/2) = -2⁷

The first appears in the Chudnovsky brothers’ pi formula,

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

e^π√163 ≈ 640320³ + 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,

e^π/2√148 = e^π√37 ≈ 14112² + 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,

e^π/3√267 = e^π√(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 T_1A, T_2A, T_3A, T_4A, 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). There are 22 that are complex quadratic valued, so is reduced to 194 - 22 = 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.)

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.) 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,

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,

T_4- + T_4|2 = 2T_8-

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,

T_4C(q) = 1/q + 20q - 62q³ + 216q⁵ - 641q⁷ + … (A007248)

T_4D(q) = 1/q - 12q + 66q³ - 232q⁵ + 639q⁷ + … (A007249)

T_8E(q) = 1/q + 4q + 2q³ - 8q⁵ - q⁷ + … (A029841)

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

T_4C + T_4D = 2T_8E

though it is proven by Atkin that the equality holds for all terms. Thus, just as one can consider the series T_27B as redundant being equal to T_27A, 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 T_pA in Ramanujan-type pi formulas? Ramanujan’s four kinds of pi formulas are of the form,

where h_p is one of various products of three Pochhammer symbols (their factorial equivalents are given in this article) and C is either j(τ), r_2A(τ), r_3A(τ), r_4A(τ) or equivalently, the McKay-Thompson series T_1A, T_2A, T_3A, T_4A 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, T_7A as well.

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

where g_p 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/π^m” for m > 1 for a list.

IV. A Continued Fraction Using e^π√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 r_4A(n) given in the Introduction and we find that it is an algebraic number,

r_4A((1/2)√-163) = x²⁴

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

R = e^π√163 ≈ 640320³ + 743.9999999999992…

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

and,

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