The jfunction and Its Cousins
(Or why e^{π√58} is also close to an integer)
by Tito Piezas III
Abstract: The McKayThompson series for the Monster group, along with certain Dedekind eta quotients, are used in a qexpansion to explain why transcendental constants of form e^{π√d} with class number h(d) ≠ 1 are also close to integers.
I. Introduction II. The jfunction, j(τ) III. The Dedekind eta function, η(τ) IV. The rfunctions r_{p}(τ) V. The eta quotients f_{p} VI. Examples and Mathematica Computations VII. Close Approximations VIII. Conclusion: Pi formulas
I. Introduction
The approximations,
e^{π√43} ≈ 12^{3}(9^{2}1)^{3} + 743.9997… e^{π√67} ≈ 12^{3}(21^{2}1)^{3} + 743.999998… e^{π√163} ≈ 12^{3}(231^{2}1)^{3} + 743.9999999999992…
are well known, the result of a beautiful theory involving complex multiplication and the modular function called the jfunction. These are the three largest Heegner numbers and have class number h(d) = 1. The pattern continues for some discriminants d with h(d) = 2, with the highest being,
e^{π√427} ≈ 12^{3}(x^{2}1)^{3} + 743.99999999999999999999998…, where x = 3(2405+308√61)
(note that d = 427 = 7*61), and so on for d with higher class numbers h(d) = n involving algebraic integers of degree n. (As long as d ≠ 3m or d ≠ 12n1.) However, how do we explain that there are others that are also close to integers,
e^{π√37} ≈ 14112^{2} + 103.99997… e^{π√10} ≈ 12^{4} 104.2… e^{π√58} ≈ 396^{4} 104.0000001…
where the “excess” hovers not around 744, but 104? Furthermore, given the fundamental unit U_{d},
U_{37} = 6+√37 U_{5} = (1+√5)/2 U_{29} = (5+√29)/2
which is involved in Pell equations, x^{2}dy^{2} = {±1, ±4}, then,
2^{6}(U_{37}^{3} U_{37}^{(3)})^{2} = 14112^{2} 2^{6}(U_{5}^{6}+ U_{5}^{(6)})^{2} = 12^{4} 2^{6}(U_{29}^{6}+ U_{29}^{(6)})^{2} = 394^{4}
The answer is an intriguing elliptic function called the Dedekind eta function.
II. The jfunction, j(τ)
Before going into the Dedekind eta function, first, the basics of the jfunction. This is a modular form of weight zero (equivalently, a modular function) defined by,
j(τ)_{ }= 1/q + 744 + 196884q + 21493760q^{2} + 864299970q^{3} + … (A000521)
where q = e^{2πiτ} = exp(2πiτ), and τ is the halfperiod ratio. This can be calculated in Mathematica as,
j(τ) = N[12^{3}KleinInvariantJ[τ], n]
for arbitrary n decimal places. We can distinguish two forms of τ. Let d be some positive integer,
Case 1. τ = (1+√d)/2 (Associated with odd discriminant d)
q = e^{2πi τ} = e^{2πi (1+√d)/2} = e^{πi (1+√d)} = e^{πi} e^{πi√d} = (1) e^{π√d} = 1/(e^{π√d})
Case 2. τ = √m (Associated with even discriminant d = 4m)
q = e^{2πi τ} = e^{2πi√m} = e^{2π√m} = 1/(e^{2π√m})
Thus as d increases, q becomes a very small real number, negative for the first case, and positive for the second. For the first, we have q = 1/(e^{π√d}) so,
j(τ)_{ }= e^{π√d} + 744 + 196884q + 21493760q^{2} + 864299970q^{3} + …
But since q is very small, the contribution of subsequent terms rapidly falls off. To illustrate for d = 163, the sum of the first n terms are,
2 terms = 640320^{3} + 7 x 10^{13} 3 terms = 640320^{3}  3 x 10^{28} 4 terms = 640320^{3} + 4 x 10^{44}
One can see the contribution of the other terms is just getting smaller and smaller. Thus, we can effectively truncate the jfunction as,
j(τ)_{ }≈ e^{π√d} + 744,
or equivalently,
e^{π√d} _{ }≈ j(τ)_{ }+ 744 It is wellestablished that,
Theorem: “Let d be a positive integer. Then j(τ) is an algebraic integer of degree n = h(d), where h(d) is the class number of d.”
If d has class number h(d) = 1, then j(τ) is an integer and explains, especially for the larger d = {43, 67, 163}, why e^{π√d} is so close to an integer. Similarly, for the second case,
j(τ)_{ }≈ e^{2π√m} + 744
so the fact that j(√7) = 255^{3} explains,
e^{π√28} ≈ 255^{3}  744.01…
though, while d = 28 has class number 1, it is not a fundamental discriminant. Both can be calculated in Mathematica for, say, n = 1000 decimal places as,
j(τ)_{ }= N[1728KleinInvariantJ[τ] /. τ → (1+√163)/2, 1000] = 640320^{3} j(τ)_{ }= N[1728KleinInvariantJ[τ] /. τ → √7, 1000] = 255^{3}
Taking note that a functional equation of the jfunction is,
j(τ) = j(τ+1)
then for τ = (1+√d)/2 or τ = √m, with d or m as positive integers, there are exactly 13 points such that j(τ) is an integer: 9 fundamental discriminants and 4 nonfundamental ones:
j(√1) = 12^{3} j(√2) = 20^{3} j((1+√3)/2) = 0 j((1+√7)/2) = 15^{3} j((1+√11)/2) = 32^{3} j((1+√19)/2) = 96^{3} j((1+√43)/2) = 960^{3} j((1+√67)/2) = 5280^{3} j((1+√163)/2) = 640320^{3}
with the first two corresponding to even discriminants d = {4, 8}. And,
j(√3) = 2*30^{3} j(√4) = 66^{3} j(√7) = 255^{3} j((1+√27)/2) = 3*160^{3}
with the first three corresponding to even discriminants d = {12, 16, 28}.
III. The Dedekind eta function, η(τ)
Just like the jfunction, the Dedekind eta function is also a modular form, but it is of weight ½. (Note: My thanks to Michael Somos for pointing out an error in a previous draft). It was introduced in 1877 by Wilhelm Dedekind (18311916) and is usually defined in terms of the infinite product,
where q = e^{2πiτ} = exp(2πiτ). As one can see, it involves a 24th root (which “explains” why the integer 24 will play such a significant role later). This can be calculated in Mathematica as,
η(τ) = N[DedekindEta[τ], n]
for arbitrary n decimal places. A remarkable feature of the η(τ) is that while for complex τ it generally yields a transcendental number, certain eta quotients turn out to be algebraic numbers. Furthermore, one can express the jfunction j(τ) discussed in the previous section using the simple formula,
j(τ)_{ }= (f_{2}^{24} + 16)^{3} / f_{2}^{24}
where f_{2} = η[τ/2]/η[τ].
The eta quotient f_{2} is one of the Weber functions. (Unfortunately, Weber labeled it as f_{1} though why it is labeled f_{2} in this paper will become clearer later.) There are in fact several other formulas using various eta quotients, but that will be for another article. What we’ll focus on is how some eta quotients can be used to define other interesting modular forms similar to j(τ).
IV. The rfunctions, r_{p}(τ)
Ramanujan did preliminary work on these functions which can be defined in terms of eta quotients. A lot of his results were on the orders p = 2,3,5,7, but there are several more. As was pointed out, the Dedekind eta function involved a 24th root. Amazingly, the orders p of the rfunctions r_{p}(τ) depend on simple arithmetic properties of the integer 24, namely, they are the positive integers p such that,
k = 24/(p1)
is also an integer. It is easily seen there are eight, p = {2, 3, 4, 5, 7, 9, 13, 25}, and can be divided into 2 kinds:
a) p ≠ 1 mod 4: p = {2, 3, 4, 7} b) p = 1 mod 4: p = {5, 9, 13, 25}
To recall, the jfunction can be given the formula,
j(τ)_{ }= (f_{2}^{24} + 16)^{3} / f_{2}^{24} = 1/q + 744 + 196884q + 21493760q^{2} + 864299970q^{3} + … (A000521, A007240)
Excluding the constant term 744, this sequence of coefficients (given by the second OEIS link) is the McKayThompson series of class 1A for the Monster group. The rfunctions, r_{p}(τ) of order p, have the similar form,
I. p ≠ 1 mod 4; p = {2, 3, 4, 7}
r_{2}(τ) = (f_{2}^{24} + 64)^{2} / f_{2}^{24} = 1/q + 104 + 4372q + 96256q^{2} + 1240002q^{3} + … (A007267)
r_{3}(τ) = (f_{3}^{12} + 27)^{2} / f_{3}^{12} = 1/q + 42 + 783q + 8672q^{2} + 65367q^{3} + … (A030197)
r_{4}(τ) = (f_{4}^{8} + 16)^{2} / f_{4}^{8} = 1/q + 24 + 276q + 2048q^{2} + 11202q^{3} + … (A097340, A107080)
r_{7}(τ) = (f_{7}^{4} + 7)^{2} / f_{7}^{4} = 1/q + 10 + 51q + 204q^{2} + 681q^{3} + … (A030183)
And no wonder, as the sequences are, for p = {2, 3, 4, 7}, the McKayThompson series of class pA for the Monster group! These functions r_{p}(τ), as eta quotients f_{p}, have a common form,
r_{p}(τ) = (f_{p}^{k} + p^{k/4})^{2} / f_{p}^{k}
where k = 24/(p1), hence k = {24, 12, 8, 4}. Interestingly, the constant term c = {104, 42, 24, 10} also has a common form,
c = 2p^{k/4}k
but I haven’t been able to figure out if the subsequent terms can be expressed in {p, k}. The other set of p are,
II. p = 1 mod 4; p = {5, 9, 13, 25}
r_{5}(τ) = (f_{5}^{12} + 125) / f_{5}^{6} = 1/q  6 + 134q + 760q^{2} + 3345q^{3} + … (A007251)
r_{9}(τ) = (f_{9}^{6} + 27) / f_{9}^{3} = 1/q  3 + 27q + 86q^{2} + 243q^{3} + … (A07266)
r_{13}(τ) = (f_{13}^{4} + 13) / f_{13}^{2} = 1/q  2 + 12q + 28q^{2} + 66q^{3} + … (A034318)
r_{25}(τ) = (f_{25}^{2} + 5) / f_{25} = 1/q  1 + 4q + 5q^{2} + 10q^{3} + … (A058594)
The sequences are the McKayThompson series of class pA for the Monster group for p = {5, 9, 13, 25}. Let k = 24/(p1), and we get k = {6, 3, 2, 1}. These functions also have a common form,
r_{p}(τ) = (f_{p}^{2k} + p^{k/2}) / f_{p}^{k}
where f_{p} as usual are eta quotients. Notice the subtle differences from the first set, like the fact that the numerator is no longer squared. The constant term c = {6, 3, 2, 1} are just k.
V. The Eta Quotients, f_{p}
For the purposes of this paper, let the qexpansion in the previous section involve q = exp(2πiτ/p). Then the eta quotients f_{p} are,
A. Even order p
f_{2} = η[τ/2] / η[τ], where τ = √m or τ = 1+√m f_{4} = η[τ/4] / η[τ], where τ = √m, or τ = 6+√m
B. Odd order p = 2b+1
Case 1: If τ = √m, then f_{p} = η[τ/p] / η[τ]. Case 2: If τ = (1+√d)/2, then f_{p} = ω_{k} η[(τ+b)/p] / η[τ].
where ω_{k} = (1)^{1/k} = exp(πi/k), and k = 24/(p1).
It is easy to evaluate these eta quotients using Mathematica. An interesting feature of f_{p} is that its kth power where k = 24/(p1) also has a qexpansion with integer coefficients. Let f_{p} = f_{p}(τ), then,
f_{2}(τ)^{24} = 1/q  24 + 276q  2048q^{2} + 11202q^{3}  49152q^{4} + … (A007191) f_{3}(τ)^{12} = 1/q  12 + 54q  76q^{2}  243q^{3} + 1188q^{4} + … (A030182)
f_{9}(τ)^{3} = 1/q  3 + 5q^{2}  7q^{5} + 3q^{8} + 15q^{11} + … (A058091)
f_{13}(τ)^{2} = 1/q  2 + q + 2q^{2} + q^{3} + 2q^{4}  2q^{5} + … (A058496)
f_{25}(τ) = 1/q  1  q + q^{4} + q^{6}  q^{11}  q^{14} + … (A096563)
And what are these sequences when normalized without the constant term? These are mostly the McKayThompson series of class pB for Monster! (With the exception of f_{4} which is 4C and f_{25} which is 25a.)
VI. Examples and Mathematica Computations
A few examples will be given to highlight the differences when using forms τ = √m, τ = a+√m (a an integer), or τ = (1+√d)/2.
A. Even order p
r_{2}(τ) = N[(f_{2}^{24} + 64)^{2} / f_{2}^{24} /. f_{2} → DedekindEta[τ/2]/DedekindEta[τ] /. τ → √58, 1000] = 396^{4} r_{2}(τ) = N[(f_{2}^{24} + 64)^{2} / f_{2}^{24} /. f_{2} → DedekindEta[τ/2]/DedekindEta[τ] /. τ → 1+√37, 1000] = 14112^{2}
B. Odd prime order p
r_{3}(τ) = N[(f_{3}^{12} + 27)^{2} / f_{3}^{12} /. f_{3} → (1)^{1/12}DedekindEta[(τ+1)/3]/DedekindEta[τ] /. τ → (1+√267)/2, 1000] = 300^{3} r_{5}(τ) = N[(f_{5}^{12} + 125) / f_{5}^{6} /. f_{5} → (1)^{1/6}DedekindEta[(τ+2)/5]/DedekindEta[τ] /. τ → (1+√235)/2, 1000] = 5^{3}(122) r_{7}(τ) = N[(f_{7}^{4} + 7)^{2} / f_{7}^{4} /. f_{7} → (1)^{1/4}DedekindEta[(τ+3)/7]/DedekindEta[τ] /. τ → (1+√427)/2, 1000] = 7(39^{2}) r_{13}(τ) = N[(f_{13}^{4} + 13) / f_{13}^{2} /. f_{13} → (1)^{1/2}DedekindEta[(τ+6)/13]/DedekindEta[τ] /. τ → (1+√403)/2, 1000] = 13(10)
These d have class number h(d) = 2 and there are exactly 18 fundamental discriminants that have it,
d = {15, 20, 24, 35, 40, 51, 52, 88, 91, 115, 123, 148, 187, 232, 235, 267, 403, 427}
With the exception of d = 187 = 11(17), all are divisible by any of the primes p = {2, 3, 5, 7, 13}. Using appropriate p, these yield integer r_{p}(τ),
r_{2}(√6) = 48^{2} r_{2}(√10) = 12^{4} r_{2}(√22) = 1584^{2} r_{2}(√58) = 396^{4}
r_{2}(1+√5) = 32^{2} r_{2}(1+√13) = 288^{2} r_{2} (1+√37) = 14112^{2}
r_{3}((1+√15)/2) = 3^{3} r_{3}((1+√51)/2) = 12^{3} r_{3}((1+√123)/2) = 48^{3} r_{3}((1+√267)/2) = 300^{3}
r_{5}((1+√35)/2) = 5^{2}(2) r_{5}((1+√115)/2) = 5^{2}(34) r_{5}((1+√235)/2) = 5^{3}(122)
r_{7}((1+√91)/2) = 7(3^{2}) r_{7}((1+√427)/2) = 7(39^{2})
r_{13}((1+√403)/2) = 13(10)
Testing it with fundamental discriminants with class number h(d) = 4, the appropriate r_{p}(τ) were algebraic integers of degree 2. For h(d) = 6, they were of degree 3. There seems to be a pattern,
Conjecture: “Given primes p = {2, 3, 5, 7, 13} and a composite fundamental discriminant d = pv with class number h(d) = 2n. Then, for appropriately chosen τ, r_{p}(τ) is an algebraic integer of degree n.”
1. For odd d: let τ = (1+√d)/2
2. For even d = 4m:
a. odd m, let τ = (1+√m)/2; (But τ = 1+√m for p = 2.)
b. even m, let τ = √m
VII. Close Approximations
The largest of the rfunctions r_{p}(τ) in the previous section give rise to close integer approximations to transcendental numbers of form e^{(π/p)√d} , namely,
e^{(π/2)√148} = e^{π√37} ≈ 14112^{2} + 103.99997… e^{(π/2)√232} = e^{π√58} ≈ 396^{4}  104.0000001…
e^{(π/3)√123} = e^{π√(41/3)} ≈ 48^{3} + 41.992… e^{(π/3)√267} = e^{π√(89/3)} ≈ 300^{3} + 41.99997…
e^{(π/5)√235} = e^{π√(47/5)} ≈ 5^{3}(122)  6.008…
e^{(π/7)√427} = e^{π√(61/7)} ≈ 7(39^{2}) + 9.995…
The reason why is analogous to the one for the jfunction. To recall, the qexpansion of r_{p}(τ) for p = 2 is,
r_{2}(τ) = 1/q + 104 + 4372q + 96256q^{2} + 1240002q^{3} + …
For τ = √m, then,
q = e^{2πi τ/p} = e^{πi√m} = e^{π√m} = 1/(e^{π√m})
As before, as m increases, q is a very small real number and the contribution of subsequent terms of the qexpansion becomes minuscule. To illustrate, for τ = √58, we saw that the formula using the eta quotients yields an integer value as r_{2}(√58) = 396^{4}. But using the qexpansion, the sum of the first n terms are,
2 terms = 396^{4}  1.7 x 10^{7} 3 terms = 396^{4}  1.6 x 10^{16} 4 terms = 396^{4}  8.3 x 10^{26}
where the difference is getting smaller and smaller. Hence, we can just truncate r_{2}(τ) as,
r_{2}(τ) ≈ 1/q + 104 ≈ e^{π√m} + 104
explaining why,
e^{π√58} ≈ 396^{4}  104.0000001…
and, by analogy,
e^{π√(89/3)} ≈ 300^{3} + 41.99997…
as well as the other examples, are so close to an integer.
VIII. Conclusion: Pi Formulas
Before concluding this article, there are many other surprising aspects to the functions j(τ) and r_{p}(τ), which I plan to write about in a series of articles. The next one involves their role in pi formulas. For example, define the factorial quotients,
h_{2} = (4n)! / (n!^{4}) h_{3} = (2n)!(3n)! / (n!^{5})
Then, for n = 0 to ∞,
1/π = 32√2 Σ h_{2} (26390n+1103)/(396^{4})^{n+1/2} 1/π = 2 Σ h_{3} (1)^{n }(14151n+827)/(300^{3})^{n+1/2}
where the blue numbers are the exact values of r_{2}(√58) and r_{3}((1+√267)/2), respectively. It turns out that r_{2}(τ) for certain τ may involve familiar constants like the golden ratio, tribonacci constant, and the plastic constant, all of which can be used in pi formulas!
But that’s another story.
 END  P.S. One of my favorite books. Trust me, you'll want to read it again and again.
© Sept 2010, Tito Piezas III
You can email author at tpiezas@gmail.com.
