0011: Article 1 (The j-function)

 
The j-function and Its Cousins

(Or why eπ√58 is also close to an integer)

 

by Tito Piezas III

 

 

Abstract:  The McKay-Thompson series for the Monster group, along with certain Dedekind eta quotients, are used in a q-expansion to explain why  transcendental constants of form eπ√d with class number h(-d) ≠ 1 are also close to integers.

 

 

I. Introduction

II. The j-function, j(τ)

III. The Dedekind eta function, η(τ)

IV. The r-functions rp(τ)

V. The eta quotients fp

VI. Examples and Mathematica Computations

VII. Close Approximations

VIII. Conclusion: Pi formulas

 

 

I. Introduction

 

The approximations,

 

eπ√43 ≈ 123(92-1)3 + 743.9997…

eπ√67 ≈ 123(212-1)3 + 743.999998…

eπ√163 ≈ 123(2312-1)3 + 743.9999999999992…

 

are well known, the result of a beautiful theory involving complex multiplication and the modular function called the j-function.  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,

 

eπ√427 ≈ 123(x2-1)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 ≠ 12n-1.)  However, how do we explain that there are others that are also close to integers,

 

eπ√37141122 + 103.99997…

eπ√10124 -104.2…

eπ√583964 -104.0000001…

 

where the “excess” hovers not around 744, but 104?  Furthermore, given the fundamental unit Ud,

 

U37 = 6+√37

U5 = (1+√5)/2

U29 = (5+√29)/2

 

which is involved in Pell equations, x2-dy2 = {±1, ±4}, then,

 

26(U373- U37(-3))2 = 141122

26(U56+ U5(-6))2 = 124

26(U296+ U29(-6))2 = 3944

 

The answer is an intriguing elliptic function called the Dedekind eta function.  

 

 

II. The j-function, j(τ)

 

Before going into the Dedekind eta function, first, the basics of the j-function.  This is a modular form of weight zero (equivalently, a modular function) defined by,

 

j(τ) = 1/q + 744 + 196884q + 21493760q2 + 864299970q3 + …   (A000521)

 

where q = e2πiτ = exp(2πiτ), and τ is the half-period ratio.  This can be calculated in Mathematica as,

 

j(τ) = N[123KleinInvariantJ[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 = e2πi τ = e2πi (1+√-d)/2 = eπi (1+√-d) = eπi eπi√-d = (-1) e-π√d = -1/(eπ√d)

 

Case 2.  τ = √-m      (Associated with even discriminant d = 4m) 

 

q = e2πi τ = e2πi√-m = e-2π√m = 1/(e2π√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/(eπ√d) so,

 

j(τ) = -eπ√d + 744 + 196884q + 21493760q2 + 864299970q3 + …

 

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 = -6403203 + 7 x 10-13

3 terms = -6403203 - 3 x 10-28

4 terms = -6403203 + 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 j-function as,

 

j(τ) ≈ -eπ√d + 744,

 

or equivalently,

 

eπ√d  ≈ -j(τ) + 744

 
It is well-established 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 eπ√d is so close to an integer.  Similarly, for the second case,

 

j(τ) ≈ e2π√m + 744

 

so the fact that j(√-7) = 2553 explains,

 

eπ√28 ≈ 2553 - 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] = -6403203

j(τ) = N[1728KleinInvariantJ[τ] /. τ → √-7, 1000] = 2553

 

Taking note that a functional equation of the j-function 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 non-fundamental ones:

 

j(√-1) = 123

j(√-2) = 203

j((1+√-3)/2) = 0

j((1+√-7)/2) = -153

j((1+√-11)/2) = -323

j((1+√-19)/2) = -963

j((1+√-43)/2) = -9603

j((1+√-67)/2) = -52803

j((1+√-163)/2) = -6403203

 

with the first two corresponding to even discriminants d = {4, 8}.  And,

 

j(√-3) = 2*303

j(√-4) = 663

j(√-7) = 2553

j((1+√-27)/2) = -3*1603

 

with the first three corresponding to even discriminants d = {12, 16, 28}.

 

 

III. The Dedekind eta function, η(τ)

 

Just like the j-function, 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 (1831-1916) and is usually defined in terms of the infinite product,
 

 

where q = e2πiτ = 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, rp(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 rp(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, rp(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 rp(τ), as eta quotients fp, have a common form,

 

rp(τ) = (fpk + pk/4)2 / fpk 

 

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 = 2pk/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,

 

rp(τ) = (fp2k + pk/2) / fpk 

 

where fp 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, fp

 

For the purposes of this paper, let the q-expansion in the previous section involve q = exp(2πiτ/p).  Then the eta quotients fp 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 fp = η[τ/p] / η[τ].

 
Case 2:  If τ = (1+√-d)/2, then fp = ωk η[(τ+b)/p] / η[τ].

 

where ωk = (-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 fp is that its kth power where k = 24/(p-1) also has a q-expansion with integer coefficients.  Let fp = fp(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 rp(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 rp(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, rp(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 rp(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 = eπ√37 ≈ 141122 + 103.99997…

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

 

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

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

 

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

 

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

 

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

 

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

 

For τ = √-m, then,

 

q = e2πi τ/p = eπi√-m = e-π√m = 1/(eπ√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 ≈ eπ√m + 104

 

explaining why,

 

eπ√583964 - 104.0000001…

 

and, by analogy,

 

eπ√(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 rp(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. 
  
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