0011: Article 1 (The j-function)

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 j-function j(τ) discussed in the previous section using the simple formula,

j(τ) = (f₂²⁴ + 16)³ / f₂²⁴

where f₂ = η[τ/2]/η[τ].

The eta quotient f₂ is one of the Weber functions. (Unfortunately, Weber labeled it as f₁ though why it is labeled f₂ 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 r-functions, 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 r-functions r_p(τ) depend on simple arithmetic properties of the integer 24, namely, they are the positive integers p such that,

k = 24/(p-1)

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 j-function can be given the formula,

j(τ) = (f₂²⁴ + 16)³ / f₂²⁴ = 1/q + 744 + 196884q + 21493760q² + 864299970q³ + … (A000521, A007240)

Excluding the constant term 744, this sequence of coefficients (given by the second OEIS link) is the McKay-Thompson series of class 1A for the Monster group. The r-functions, r_p(τ) of order p, have the similar form,

I. p ≠ 1 mod 4; p = {2, 3, 4, 7}

r₂(τ) = (f₂²⁴ + 64)² / f₂²⁴ = 1/q + 104 + 4372q + 96256q² + 1240002q³ + … (A007267)

r₃(τ) = (f₃¹² + 27)² / f₃¹² = 1/q + 42 + 783q + 8672q² + 65367q³ + … (A030197)

r₄(τ) = (f₄⁸ + 16)² / f₄⁸ = 1/q + 24 + 276q + 2048q² + 11202q³ + … (A097340, A107080)

r₇(τ) = (f₇⁴ + 7)² / f₇⁴ = 1/q + 10 + 51q + 204q² + 681q³ + … (A030183)

And no wonder, as the sequences are, for p = {2, 3, 4, 7}, the McKay-Thompson 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/(p-1), 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}

r₅(τ) = (f₅¹² + 125) / f₅⁶ = 1/q - 6 + 134q + 760q² + 3345q³ + … (A007251)

r₉(τ) = (f₉⁶ + 27) / f₉³ = 1/q - 3 + 27q + 86q² + 243q³ + … (A07266)

r₁₃(τ) = (f₁₃⁴ + 13) / f₁₃² = 1/q - 2 + 12q + 28q² + 66q³ + … (A034318)

r₂₅(τ) = (f₂₅² + 5) / f₂₅ = 1/q - 1 + 4q + 5q² + 10q³ + … (A058594)

The sequences are the McKay-Thompson series of class pA for the Monster group for p = {5, 9, 13, 25}. Let k = 24/(p-1), 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 q-expansion in the previous section involve q = exp(2πiτ/p). Then the eta quotients f_p are,

A. Even order p

f₂ = η[τ/2] / η[τ], where τ = √-m or τ = 1+√-m

f₄ = 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/(p-1).

It is easy to evaluate these eta quotients using Mathematica. An interesting feature of f_p is that its kth power where k = 24/(p-1) also has a q-expansion with integer coefficients. Let f_p = f_p(τ), then,

f₂(τ)²⁴ = 1/q - 24 + 276q - 2048q² + 11202q³ - 49152q⁴ + … (A007191)

f₃(τ)¹² = 1/q - 12 + 54q - 76q² - 243q³ + 1188q⁴ + … (A030182)

f₄(τ)⁸ = 1/q - 8 + 20q - 62q³ + 216q⁵ - 641q⁷ + … (A007248)

f₅(τ)⁶ = 1/q - 6 + 9q + 10q² - 30q³ + 6q⁴ + … (A007252)

f₇(τ)⁴ = 1/q - 4 + 2q + 8q² - 5q³ - 4q⁴ - 10q⁵ + …(A052240)

f₉(τ)³ = 1/q - 3 + 5q² - 7q⁵ + 3q⁸ + 15q¹¹ + … (A058091)

f₁₃(τ)² = 1/q - 2 + q + 2q² + q³ + 2q⁴ - 2q⁵ + … (A058496)

f₂₅(τ) = 1/q - 1 - q + q⁴ + q⁶ - q¹¹ - q¹⁴ + … (A096563)

And what are these sequences when normalized without the constant term? These are mostly the McKay-Thompson series of class pB for Monster! (With the exception of f₄ which is 4C and f₂₅ 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

r₂(τ) = N[(f₂²⁴ + 64)² / f₂²⁴ /. f₂ → DedekindEta[τ/2]/DedekindEta[τ] /. τ → √-58, 1000] = 396⁴

r₂(τ) = N[(f₂²⁴ + 64)² / f₂²⁴ /. f₂ → DedekindEta[τ/2]/DedekindEta[τ] /. τ → 1+√-37, 1000] = -14112²

B. Odd prime order p

r₃(τ) = N[(f₃¹² + 27)² / f₃¹² /. f₃ → (-1)^1/12DedekindEta[(τ+1)/3]/DedekindEta[τ] /. τ → (1+√-267)/2, 1000] = -300³

r₅(τ) = N[(f₅¹² + 125) / f₅⁶ /. f₅ → (-1)^1/6DedekindEta[(τ+2)/5]/DedekindEta[τ] /. τ → (1+√-235)/2, 1000] = -5³(122)

r₇(τ) = N[(f₇⁴ + 7)² / f₇⁴ /. f₇ → (-1)^1/4DedekindEta[(τ+3)/7]/DedekindEta[τ] /. τ → (1+√-427)/2, 1000] = -7(39²)

r₁₃(τ) = N[(f₁₃⁴ + 13) / f₁₃² /. f₁₃ → (-1)^1/2DedekindEta[(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),

r₂(√-6) = 48²

r₂(√-10) = 12⁴

r₂(√-22) = 1584²

r₂(√-58) = 396⁴

r₂(1+√-5) = -32²

r₂(1+√-13) = -288²

r₂ (1+√-37) = -14112²

r₃((1+√-15)/2) = -3³

r₃((1+√-51)/2) = -12³

r₃((1+√-123)/2) = -48³

r₃((1+√-267)/2) = -300³

r₅((1+√-35)/2) = -5²(2)

r₅((1+√-115)/2) = -5²(34)

r₅((1+√-235)/2) = -5³(122)

r₇((1+√-91)/2) = -7(3²)

r₇((1+√-427)/2) = -7(39²)

r₁₃((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 r-functions 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 ≈ 14112² + 103.99997…

e^(π/2)√232 = e^π√58 ≈ 396⁴ - 104.0000001…

e^(π/3)√123 = e^π√(41/3) ≈ 48³ + 41.992…

e^(π/3)√267 = e^π√(89/3) ≈ 300³ + 41.99997…

e^(π/5)√235 = e^π√(47/5) ≈ 5³(122) - 6.008…

e^(π/7)√427 = e^π√(61/7) ≈ 7(39²) + 9.995…

The reason why is analogous to the one for the j-function. To recall, the q-expansion of r_p(τ) for p = 2 is,

r₂(τ) = 1/q + 104 + 4372q + 96256q² + 1240002q³ + …

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 q-expansion becomes minuscule. To illustrate, for τ = √-58, we saw that the formula using the eta quotients yields an integer value as r₂(√-58) = 396⁴. But using the q-expansion, the sum of the first n terms are,

2 terms = 396⁴ - 1.7 x 10^-7

3 terms = 396⁴ - 1.6 x 10^-16

4 terms = 396⁴ - 8.3 x 10^-26

where the difference is getting smaller and smaller. Hence, we can just truncate r₂(τ) as,

r₂(τ) ≈ 1/q + 104 ≈ e^π√m + 104

explaining why,

e^π√58 ≈ 396⁴ - 104.0000001…

and, by analogy,

e^π√(89/3) ≈ 300³ + 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,

h₂ = (4n)! / (n!⁴)

h₃ = (2n)!(3n)! / (n!⁵)

Then, for n = 0 to ∞,

1/π = 32√2 Σ h₂ (26390n+1103)/(396⁴)^n+1/2

1/π = 2 Σ h₃ (-1)ⁿ (14151n+827)/(300³)^n+1/2

where the blue numbers are the exact values of r₂(√-58) and r₃((1+√-267)/2), respectively. It turns out that r₂(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.

◄ Previous Page Next Page ►