Part 2: PrimeGenerating Polynomials and 43 Moonshine Functions
By Tito Piezas III
Keywords: moonshine functions, Monster group, prime generating polynomial, McKayThompson series, class numbers, 43, 163.
Abstract: An expository work on how 1) primegenerating polynomials are connected to integer values of some moonshine functions r_{p}(τ); and 2) for p > 1 satisfying certain constraints, it is conjectured exactly 43 of the moonshine functions obey certain rules regarding class numbers.
I. Introduction II. Class number h(d) = 1 III. Class number h(d) = 2n IV. Class number h(d) = 4n V. Class number h(d) = 8n VI. Conjectures VII. RogersRamanujan Continued Fraction
I. Introduction
In “Part 1: The 163 Dimensions of the Moonshine Functions”, it was discussed how Conway, Norton, and Atkin showed that the moonshine functions curiously span a linear space of dimension 163. (For details, it is recommended one reads Part 1 first.) These functions assume the form of McKayThompson series,
T_{p}(q) = 1/q + a_{0} + a_{1}q + a_{2}q^{2} + a_{3}q^{2} + …
where (throughout this article),
q = e^{2πiτ} = exp(2πiτ)
the powers of q are either consecutive or some other arithmetic progression, and a_{0} is normalized as zero or another integer. But other than having a dimension of 163, another interesting aspect to them is the values they assume for τ a complex root of a quadratic, especially if this quadratic happens to be a maximal primegenerating polynomial. For example, define r_{2A}(τ) as T_{2A} with a_{0} = 104, hence,
r_{2A}(τ) = 1/q + 104 + 4372q + 96256q^{2} + 1240002q^{3} + … (A101558)
Then consider the primegenerating polynomial,
P(n) = 2n^{2}+29
which is distinctly prime for its maximum possible range of n = {0 to 28}. Solving for P(n) = 0, hence n = τ = (1/2)√58 and plugging this into the series, one finds that,
r_{2A}((1/2)√58) = 396^{4}
This particular integer can be expressed in terms of the fundamental unit U_{29} = (5+√29)/2 as,
2^{6}(U_{29}^{6} + 1/U_{29}^{6})^{2} = 396^{4}
and also appears in a pi formula found by Ramanujan a century ago in 1912,
where one can see the 58 as well. And, of course, in the almost integer,
e^{π√58} = 396^{4}  104.0000001…
This article, part 2 of a series, will look at an interesting section of the 163 dimensions of the moonshine functions and discuss its relationship to quadratic primegenerating polynomials. Since the class number h(d) of our example d = 4∙58 = 232 is h(d) = 2, we will start with h(d) = 1.
II. Class number h(d) = 1
Using T_{1A} with a_{0} = 744, we have the classical jfunction j(τ),
j(τ) = 1/q + 744 + 196884q + 21493760q^{2} + 864299970q^{3} + … (A007240)
There are exactly 9 negative fundamental discriminants d such that h(d) = 1, namely,
d = {4, 8, 3, 7, 11, 19, 43, 67, 163}
(The list of d with class number up to h(d) = 25 can be found in Mathworld.) The table below shows the associated primegenerating polynomial P(n) = an^{2}+bn+c, its discriminant d, and j(τ) using a root τ of P(n) = 0 which, of course, is given by the quadratic formula,
For negative d, the root on the upper half of the complex plane is chosen per the requirements of modular forms. Hence,
Table 1
Some remarks:
1. They have a beautifully consistent “internal” structure, don’t they? The reason for the squares within the cubes is due to certain Eisenstein series (to be discussed in some other article). This pattern also carries over to higher class numbers for some d. For example, the largest d (in absolute value) with h(d) = 2 is d = 427 = 7∙61, hence letting τ = (1+√427)/2, then,
j(τ) = 12^{3}(w^{2}1)^{3}, where w = 7215+924√61
an algebraic number of deg 2 as expected, since the jfunction detects the class number of negative fundamental d. This also implies,
e^{π√427} ≈ 12^{3}(w^{2}1)^{3} + 743.9999999999999999999….
which goes on for 22 nines, more than the 12 nines of d = 163. Later it will be seen that the moonshine functions, in a special way, can also detect h(d).
2. The P(n) = an^{2}+bn+c in Table 1 for c > 1 all have the maximum possible range for distinct primes as n = {1 to c1}.
3. It is not so wellknown that, starting with some negative n, then every third value of these P(n) is prime for the same length of consecutive n. And for higher d, the primes within this span are different.
For example, the sequence of 40 primes for Euler’s polynomial P(n) is {41, 43, …,1601}, while the 40 primes for P(m) will start and end differently as {6203, 5741, … ,1523}. Also, note that 2 ∙{21, 33, 81} + 1 = {43, 67, 163}. Furthermore, if we solve P(m) = 0 and plug the correct root τ into the McKayThompson series T_{9A} with constant term a_{0} = 3 (OEIS A007266),
we get complex deg 4 algebraic numbers. Given a cube root of unity, u = e^{2πi/3} = (1+√3)/2, then,
with the integers,
{j_{43}, j_{67}, j_{163}} = {960/12, 5280/12, 640320/12}
Recall that,
e^{π√43}  744 ≈ 960^{3} = 12^{3}(9^{2}1)^{3} e^{π√67}  744 ≈ 5280^{3 } = 12^{3}(21^{2}1)^{3} e^{π√163}  744 ≈ 640320^{3} = 12^{3}(231^{2}1)^{3}
Interesting that the blue numbers should appear using r_{9A}(τ) and primegenerating polynomials of form P(n) = 9n^{2}+bn+c.
III. Class number h(d) = 2n
The Monster group has the extremely large order,
O = 2^{4}∙ 3^{4}∙ 5^{4}∙ 7^{4}∙ 11^{4}∙13^{4}∙ 17∙ 19∙ 23∙ 29∙ 31∙ 41∙ 47∙ 59∙ 71 ≈ 8∙ 10^{53}
The 15 primes that divide its order are called the supersingular primes, and there is a McKayThompson series T_{pA} for each one. Most of these primes also divide the fundamental discriminants d with class number h(d) = 2. Define the following functions,
r_{2A}(τ) = 1/q + 104 + 4372q + 96256q^{2} + 1240002q^{3} + … (A101558)
r_{3A}(τ) = 1/q + 42 + 783q + 8672q^{2} + 65367q^{3} + … (A030197)
r_{5A}(τ) = 1/q  6 + 134q + 760q^{2} + 3345q^{3} + … (A007251)
r_{7A}(τ) = 1/q + 10 + 51q + 204q^{2} + 681q^{3} + … (A030183)
r_{11A}(τ) = 1/q + 0 + 17q + 46q^{2} + 116q^{3} + … (A030183)
r_{13A}(τ) = 1/q  2 + 12q + 28q^{2} + 66q^{3} + 132q^{4} + … (A034318)
and so on. (For brevity, we will focus only on the smaller p.) Without the constant term, these are simply the series T_{2A}, T_{3A}, T_{5A}, T_{7A},_{ }T_{11A}, and T_{13A}, and the first two have already been given in Part 1. There are exactly 18 negative fundamental discriminants d such that h(d) = 2,
d = {15, 20, 24, 35, 40, 51, 52, 88, 91, 115, 123, 148, 187, 232, 235, 267, 403, 427}
The table below shows their associated primegenerating polynomials P(n), and the value of r_{p}(τ) using the root τ of P(n) = 0:
Table 3
Some remarks:
1. The values are also beautifully consistent in form. While the table used class number h(d) = 2, one can also use h(d) = 4, 6, etc. For ex, let d = 4∙82 with h(d) = 4, and τ = (1/2)√82, then one gets an algebraic number just of deg 2,
r_{2A}(τ) = 12^{4}(51+8√41)^{4}
2. Using large d, one gets almostintegers (or in general, almostalgebraics),
e^{π/2√232} = e^{π√58} ≈ 396^{4}^{ } 104.0000001….
e^{π/3√267} = e^{π√(89/3)} ≈ 300^{3} + 41.99997…
e^{π/5√235} = e^{π√(47/5)} ≈ 15250  6.008…
e^{π/7√427} = e^{π√(61/7)} ≈ 7∙39^{2}^{ }+ 9.995…
e^{π/13√403} = e^{π√(31/13)} ≈ 130^{ } 2.09…
and,
e^{π/2√328} = e^{π√82} ≈ 12^{4}(51+8√41)^{4}^{ } 104.000000001…
IV. Class number h(d) = 4n
In contract to the ones in the previous sections, a primegenerating polynomial of form P(n) = 6n^{2}6n+c cannot be prime for n = c1 since it factors at that value. For P(n) = 10n^{2}10n+c, it is n = c2, and so on. The most impressive for the first kind is,
P(n) = 6n^{2}6n+31
which is distinctly prime for the maximum n = {1 to 29}. This has discriminant d = 708 = 6∙118, and we find this d also appears in an almostinteger,
e^{π/6√708} = e^{π√(118/6)} ≈ 1060^{2}^{ }+ 9.99992… ^{ } But this d has class number h(d) = 4. If 1060^{2} is the value of a moonshine function, then there must be some yielding, for appropriate d, algebraic numbers that are onefourth the class number. It turns out these apparently are the McKayThompson series of order p where p is the product of two distinct supersingular primes. There are 23 such orders,
2 ∙{3, 5, 7, 11, 13, 17, 19, 23, 31, 47} = {6, 10, 14, 22, 26, 34, 38, 46, 62, 94} 3 ∙{5, 7, 11, 13, 17, 19, 23, 29, 31} = {15, 21, 33, 39, 51, 57, 69, 87, 93} 5 ∙{7, 11, 19} = {35, 55, 95} 7 ∙{17} = 119
though p = 57 and 93 are to be excluded. (For the relevant McKayThompson series in this family, these two are the only ones where the powers of q are not consecutive but are in the progression 3m+2.) So what remains are 21 series. For brevity, we will focus only in the case p = 2m for small m. Define the following functions,
r_{6A}(τ) = 1/q + 10 + 79q + 352q^{2} + 1431q^{3} + … (A007254)
r_{10A}(τ) = 1/q + 4 + 22q + 56q^{2} + 177q^{3} + … (A058097)
r_{14A}(τ) = 1/q + 2 + 11q + 20q^{2} + 57q^{3} + … (A058497)
r_{22A}(τ) = 1/q  5 + 5q + 6q^{2} + 16q^{3} + 20q^{4} … (A058567)
r_{26A}(τ) = 1/q + 0 + 4q + 4q^{2} + 10q^{3} + 12q^{4} … (A058596)
and so on, which, needless to say, without the constant terms are the McKayThompson series T_{6A}, T_{10A}, T_{14A}, T_{22A}, and T_{26A}. There are exactly 54 negative fundamental discriminants d with h(d) = 4. However, only some are divisible by 6, 10, 14, 22, or 26. These, and their associated P(n), and r_{pA}(τ) are listed in the following table:
Table 4
An example using h(d) = 8 is d = 1380, hence τ = (6+√1380)/12, giving the deg 2 algebraic,
r_{6A}(τ) = 4^{2}(2093+540√15)^{2}
and implying,
e^{π/6√1380} = e^{π√(230/6)} ≈ 4^{2}(2093+540√15)^{2 }+ 9.9999997…
V. Class number h(d) = 8n
On a suggestion by Simon Norton who asked about p a product of three distinct supersingular primes, it seems there are indeed moonshine functions that could detect h(d) = 8n. There are 7 such p, namely p = {30, 42, 66, 70, 78, 105, 110}, thus 7 series,
r_{30B}(τ) = 1/q + 0 + 4q + 2q^{2} + 6q^{3} + … (A058613)
r_{42A}(τ) = 1/q + 0 + 2q + 2q^{2} + 3q^{3} + … (A058671)
r_{66A}(τ) = 1/q + 0 + 2q + 0q^{2} + q^{3} + … (A058739)
r_{70A}(τ) = 1/q + 0 + q + 0q^{2} + 2q^{3} + … (A058744)
r_{78A}(τ) = 1/q + 0 + q + q^{2} + q^{3} + … (A058754)
r_{105A}(τ) = 1/q + 0 + q + q^{2} + 0q^{3} + … (A058773)
r_{110A}(τ) = 1/q + 0 + 0q + q^{2} + q^{3} + … (A058774)
which, since the constant term a_{0} = 0, are simply McKayThompson series. (Note that the first uses the conjugacy class 30B.) There are exactly 131 negative fundamental d with h(d) = 8, but only some are divisible by the seven p. For brevity, we will give at most only two examples per function,
Table 5
Remarks:
1. Just like in the previous three tables, all the given primegenerating P(n) in Table 4 are distinctly prime for its maximum range n. For example, the form P(n) = 105n^{2}105n+c can only be distinctly prime for n = {1 to c24} since P(c23) will factor.
2. For class number h(d) = 16, an example is d = 5208 = 84∙62. Let τ = (1/84)√5208, and we get an algebraic number just of deg 2,
r_{42A}(τ) = (1/4)(15+√217)^{2}
VI. Conjectures
Given composite negative fundamental discriminant d = pm, but excluding d such that pd = a^{2}, or pd = a^{2}v where v divides p:
Conjecture 1: If d = pm has class number h(d) = 2n, then for p a supersingular prime, the moonshine function r_{pA}(τ) is an algebraic number of degree onehalf the h(d). There are 15 such functions.
Conjecture 2: If d = pm has class number h(d) = 4n, then for p a product of 2 distinct supersingular primes (excepting products p = 57 and 93), the r_{p}(τ) of the appropriate conjugacy class is an algebraic number of degree onefourth the h(d). There are 21 such functions.
Conjecture 3: If d = pm has class number h(d) = 8n, then for p a product of 3 distinct supersingular primes, the r_{p}(τ) of the appropriate conjugacy class is an algebraic number of degree oneeighth the h(d). There are 7 of these.
Conjecture 4: Thus, there is a total of 15 + 21 + 7 = 43 such functions.
Note 1: All the functions are from the series T_{pA}, excepting three: T_{30B}, T_{33B}, and T_{46C}.
Note 2: To recall, there are 172 McKayThompson series for the Monster, all distinct except the special case of T_{27A} = T_{27B}. And 172 = 4∙43. In fact, the Monster has at least 43 conjugacy classes of maximal subgroups. If the numbers 43, 67, 163 turn out to be significant for moonshine functions (none is yet known for the second), then that would be an interesting triple “coincidence”.
VII. RogersRamanujan Continued Fraction
In one of his letters to Hardy, Ramanujan gave the beautiful continued fraction,
We saw how supersingular primes play an important role in the moonshine functions. It turns out that simple arithmetic properties of these primes allow it to be also connected to the RogersRamanujan continued fraction. But again, that is another story…
 End 
© July 2011, Tito Piezas III You can email author at tpiezas@gmail.com
