Ramanujan’s Constant (eπ√163) And Its Cousins

 

by Titus Piezas III

 

 

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty.” – Bertrand Russell

 

Keywords: Ramanujan, e^pi, Sqrt(163), modular functions, class polynomials.

 

Contents

 

I. Introduction

II. Hilbert Class Polynomials

IIII. Weber Class Polynomials

IV. Ramanujan Class Polynomials

V. Pi Formulas

VI. Monster Group and Conclusion

 

 

I.  Introduction

 

            It is quite easy to introduce this topic such that the educated layman with an interest in mathematics can appreciate it.  Given e =  2.718281…, the base of natural logarithms, one can easily show that,

 

 eπ√163 = 262,537,412,640,768,743.99999999999925…

 

The mathematical constants e and p are transcendental numbers, that is, they can never be the roots of finite equations with coefficients in the rational field.  Yet, here we have a combination of e and p that is almost an integer.  One perhaps can assume it to be mere coincidence; of the infinity of possible numbers of the form eπ√d for some positive integer d, it may be expected there will be some that will be close to an integer.

However, when the above number, as well as others, shows a certain “internal structure”, namely,

 

eπ√67 = 52803 + 743.9999986…

eπ√163 = 6403203 + 743.99999999999925…

 

including some relations that involve square roots,

 

            eπ√22 ≈ 26(1+√2)12 + 23.99988…

            eπ√58 ≈ 26((5+√29)/2)12 + 23.999999988…

 

and,

 

            eπ√42 ≈ 44(21+8√6)4 – 104.0000062…

eπ√130 ≈ 124(323+40√65)4 – 104.0000000000012…

 

one cannot dismiss it as just coincidence.  Something interesting is going on.

It turns out the answer has to do with modular functions and what are called class polynomials, namely the Hilbert, Weber, and Ramanujan class polynomials, respectively, for the three pairs of examples above.

 

II. Hilbert Class Polynomials

 

The modular function involved in the first pair of examples is known as the j-function.  This function, j(q), has the series expansion,

 

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

 

where q = e2pit and t is the half-period ratio.  The expansion somehow “explains” why in the first pair of examples we find the approximations for 744. 

The j(q) is dependent on the t.  If t involves a quadratic irrational Öd, then an important result is that j(q) is an algebraic number of degree n, where n is the class number of d.  We can say two things: First, this algebraic number is in fact an algebraic integer, or it is defined by an equation with the leading coefficient as 1 (a monic equation). Second, this equation, the Hilbert class polynomial, is a solvable equation, or solvable using only a finite number of arithmetic operations and root extractions.  Thus, for any transcendental number eπ√d, if we know the class number n of d, then we can find an approximation to it involving an algebraic number of degree n solvable in radicals.

Class numbers are involved in the study of number fields, though we need not go into the details here.  The list of numbers d, or discriminants, belonging to particular class numbers n has been made for smaller n.  One is referred to Mathworld, http://mathworld.wolfram.com/ClassNumber.html for a listing of up to class number 25.

For class number 1, there are 9 discriminants, namely, 3, 4,7, 8, 11, 19, 43, 67, 163, also called the Heegner numbers.  The last five give the more impressive approximations to eπ√d involving integers, since j(q) would be an algebraic integer of degree 1, which is simply a plain integer.  However, there are also discriminants of class number 2, so j(q) would be an algebraic integer of degree 2, or a root of a quadratic. And so on for the higher class numbers, examples of which we will give here.

            I have tabulated the j(q) of the lower class numbers, with the entries for class numbers 1 and 2 computed by myself.  They have been grouped in a manner that will be justified eventually.  For the highest d of class number 1 and 2, I also included the error difference of the approximation.  While eπ√163 is visually impressive, instantly recognizable as a near integer and thus almost the root of a linear equation with a difference of only 7.5 x 10-13, the transcendental number eπ√427 is more numerically impressive, though how many of us can recognize at a glance that it misses being the root of a quadratic by a mere 1.3 x 10-23?

 

Class Number 1 (9 discriminants)

 

            eπ√11 ≈ 323 + 738

            eπ√19 ≈ 963 + 744

            eπ√43 ≈ 9603 + 744

            eπ√67 ≈ 52803 + 744

            eπ√163 ≈ 6403203 + 744  (7.5 x 10-13)

 

            I have included only the five highest Heegner numbers.  Technically, there are 13 discriminants with class number 1, though only 9 are maximal.  Some of the others are,

 

            eπ√16 ≈ 663  - 744

            eπ√28 ≈ 2553 - 744

 

Class Number 2 (18 discriminants)

 

            eπ√20 ≈ 23(25+13√5)3 - 744

            eπ√52 ≈ 303(31+9√13)3 - 744

            eπ√148 ≈ 603(2837+468√37)3 - 744

 

            eπ√24 ≈ 123(1+Ö2)2(5+2Ö2)3 - 744

            eπ√40 ≈ 63(65+27√5)3 - 744

            eπ√88 ≈ 603(155+108√2)3 - 744

            eπ√232 ≈ 303(140989+26163√29)3 - 744

 

            eπ√35 ≈ 163(15+7√5)3 + 744

            eπ√91 ≈ 483(227+63√13)3 + 744

            eπ√115 ≈ 483(785+351√5)3 + 744

            eπ√187 ≈ 2403(3451+837√17)3 + 744

            eπ√235 ≈ 5283(8875+3969√5)3 + 744

            eπ√403 ≈ 2403(2809615+779247√13)3 + 744

            eπ√427 ≈ 52803(236674+30303√61)3 + 744  (1.3 x 10-23)

 

            The above numbers follow the pattern of Ramanujan’s constant, approximately cubes of algebraic numbers plus 744 if the discriminant is odd, minus 744 if the discriminant is even.  (Other than eπ√24 but that’s because it involves d that is a multiple of 3.)  For the last four discriminants of class number 2, namely d = 15, 51, 123, 267, it took some time to find j(q) and it turned out that for these discriminants, j(q) is not a perfect cube.  Mathworld would give eπ√51 as,

 

            eπ√51 ≈ 4*483(6263+1519Ö17) + 744

 

However, these discriminants also happened to be multiples of 3.  I was aware of a technique (see paper by Yui and Zagier) to factor certain algebraic roots and express it in terms of fundamental units, and another factor which turns out to be a perfect cube.  So we can have expressions for the j(q) of these discriminants using smaller numbers, namely the square of a fundamental unit and a perfect cube, given below:

 

            eπ√15 ≈ 33((1+Ö5)/2)2(5+4Ö5)3 + 743

            eπ√51 ≈ 483(4+Ö17)2(5+Ö17)3 + 744

            eπ√123 ≈ 4803(32+5Ö41)2(8+Ö41)3 + 744

            eπ√267 ≈ 2403(500+53Ö89)2(625+53Ö89)3 + 744  (1.0 x 10-17)

 

Note that:

            (1/2)2 - 5*(1/2)2 = -1

            42 -17*12 = -1

            322 - 41*52 = -1

            5002 - 89*532 = -1

 

Class Number 3 (16 discriminants)

 

            For the next class numbers 3, 4, 5, we will not give the complete list but just a few examples.  Only eπ√59 and eπ√83 have been derived by myself, the others are from other authors.  The drawback of Hilbert class polynomials is the size of their coefficients and a signature of these polynomials is that their constant term is a perfect cube, which may be an indication that the root is a perfect cube of an algebraic number.  I have observed that for some discriminants, we indeed can have a simplification of these polynomials. For example, while for d = 23 the Hilbert class polynomial is given by (see paper by Morain),

 

            y3+3491750y2-5151296875y+233753 = 0

 

by setting y = x3, it will factor such that, after scaling, it simplifies to the equation below.  The other class polynomials have been reduced in the same manner.  I believe that for odd class numbers, the Hilbert class polynomial in the variable y, by setting y = x3, can be factored so that it will have smaller coefficients.  For even class numbers, especially if the discriminant is a multiple of 3, it may not simplify so easily.  We have then,

 

            eπ√23 ≈ 53x3 + 744; (x3-31x2+26x-187=0)

            eπ√31 ≈ 33x3 + 744; (x3-114x2+93x-4301=0)

            eπ√59 ≈ 323x3 + 744; (x3-98x2+67x-22=0)

            eπ√83 ≈ 1603x3 + 744; (x3-87x2+5x-2=0)

 

Class Number 4 (54 discriminants)

 

            eπ√55 ≈ x3 + 744; (x4-2355x3-8370x2-5553900x-26484975=0)

            eπ√56 ≈ 43x3 – 744; (x4-646x3+8347x2-11286x+84337=0)

 

Class Number 5 (25 discriminants)

 

            eπ√47 ≈ 53x3 + 744; (x5-264x4+484x3-15419x2+21714x-80707=0)

           

and so on…

 

III. Weber Class Polynomials

 

            The modular function involved in the second pair of examples is known as the Weber modular function.  For brevity, perhaps we can call it as the w-function or w(q).  This function has the series expansion,

 

            w(q) = 1/q + 24 + 276q + 2048q2 + 11202q3 + …

 

Again, we can see in the expansion why in the second pair of examples we have approximations to a certain constant, this time the integer 24.  Just like with the j-function j(q), the w(q) is also an algebraic number determined by an equation of degree k dependent on the class number n of the discriminant d.  This equation, the Weber class polynomial, is also solvable in radicals.

            However, there are some complications with regards to the degree since it is not necessarily k = n.  The degree k is also dependent on the nature of the discriminant d, especially for odd d if it is of the form 8m+3 or 8m+7.

 

A. Odd Class Numbers

 

Let x be the real root of its class polynomial.  If a discriminant d of form 8m+3 has class number n, then the general form is,

            eπ√d ≈ x24 - 24

where x is a root of an equation of degree 3n with a constant term -2n.  (The exception to this rule seems to be d = 3, which has a Weber class polynomial that is not a cubic.)  If a discriminant d of form 8m+7 has class number n, then the general form is,

eπ√d ≈ 212x24 – 24

where x is a root of an equation of degree n with a constant term -1. 

The Weber class polynomials for odd class numbers up to nine I found by myself  using the Integer Relations applet found at http://www.cecm.sfu.ca/projects/IntegerRelations/ .  I admit there was a certain satisfaction in finding it independently, as you use the approximations (eπ√d + 24)1/24 and (eπ√d + 24)1/24/Ö2, gradually increase the sensitivity of the applet, and it would churn out candidate polynomials with increasingly large coefficients, then suddenly there would be this polynomial with small coefficients, sometimes just single digits, with much higher accuracy than the one before and you know this was the one you were looking for.

            This was a few years back. Since then, Annegret Weng has made available the Weber class polynomials of up to d = 422500.  See http://www.exp-math.uni-essen.de/zahlentheorie/classpol/class.html .  For the list below, we will give for d of form 8m+3 only for class number one, which would then have cubic class polynomials (with the exception of d = 3).  For other odd class numbers n, it is understood that such d would have class polynomials of degree 3n, which are quite tedious to write down.  However, we can do so for d of form 8m+7.

 

Class Number 1 (9 discriminants, 6 of the form 8m+3)

 

            eπ√11 ≈ x24 – 24; (x3-2x2+2x-2=0)

            eπ√19 ≈ x24 – 24; (x3-2x-2=0)

            eπ√43 ≈ x24 – 24; (x3-2x2-2=0)

            eπ√67 ≈ x24 – 24; (x3-2x2-2x-2=0)

            eπ√163 ≈ x24 – 24; (x3-6x2+4x-2=0)

 

Class Number 3 (16 discriminants, 2 of the form 8m+7)

 

            eπ√23 ≈ 212x24 – 24; (x3-x-1=0)

            eπ√31 ≈ 212x24 – 24; (x3-x2-1=0)

 

Class Number 5 (25 discriminants, 4 of the form 8m+7)

 

            eπ√47 ≈ 212x24 – 24; (x5-x3-2x2-2x-1=0)

            eπ√79 ≈ 212x24 – 24; (x5-3x4+2x3-x2+x-1=0)

eπ√103 ≈ 212x24 – 24; (x5-x4-3x3-3x2-2x-1=0)

            eπ√127≈ 212x24 – 24; (x5-3x4-x3+2x2+x-1=0)

 

Class Number 7 (31 discriminants, 5 of the form 8m+7)

 

            eπ√71 ≈ 212x24 – 24; (x7-2x6-x5+x4+x3+x2-x-1=0)

            eπ√151 ≈ 212x24 – 24; (x7-3x6-x5-3x4-x2-x-1=0)

            eπ√223 ≈ 212x24 – 24; (x7-5x6-x4-4x3-x2-1=0)

            eπ√463 ≈ 212x24 – 24; (x7-11x6-9x5-8x4-7x3-7x2-3x-1=0)

            eπ√487 ≈ 212x24 – 24; (x7-13x6+4x5-4x4+7x3-4x2+x-1=0)

           

and so on…

 

B. Even Class Numbers

 

            While the discriminants d of odd class numbers seem to be always odd (other than class number 1 which has d = 4, 8), discriminants of even class numbers are a mix of odd and even.  I have observed that given even discriminants 4p or 8q of even class numbers with p or q prime, the appropriate root of the Weber class polynomial seems to approximate eπ√p and eπ√(2q).  We can call the first as group 1 and the second as group 2, a grouping I also used earlier.  I will be limiting this section only to these and not the odd discriminants of even class numbers.

             For the w(q) of class number 4 labeled Others, these were taken from Weber’s book (Lehrbuch der Algebra), an old book I found in the library.  These radicals are too beautiful to be locked up in the musty pages of an old book.

 

Class Number 2

 

Group 1                                                           Group 2

 

            eπ√5 ≈ 26(f)6 – 24                                 eπ√6 ≈ 26(1+√2)4 + 24

            eπ√13 ≈ 26((3+√13)/2)6 – 24                  eπ√10 ≈ 26(f)12 + 24

            eπ√37 ≈ 26(6+√37)6 – 24                        eπ√22 ≈ 26(1+√2)12 +24

eπ√58 ≈ 26((5+√29)/2)12 +24

 

where f  is the golden ratio (1+√5)/2 = 1.61803… It’s interesting how this number crops up in the expressions for w(q) whenever d is a multiple of 5, though perhaps it is to be expected since it is also a fundamental unit.  (The other 11 discriminants, other than eπ√15, are almost-roots of sextics.)

 

Class Number 4

 

Group 1                                                           Group 2

 

            eπ√17 ≈ 26(P√17)12 – 24                         eπ√14 ≈ 26(P√14)12 + 24

            eπ√73 ≈ 26(P√73)12 – 24                         eπ√34 ≈ 26(P√34)12 + 24

            eπ√97 ≈ 26(P√97)12 – 24                         eπ√46 ≈ 26(P√46)12 + 24

            eπ√193 ≈ 26(P√193)12 – 24                     eπ√82 ≈ 26(P√82)12 + 24

eπ√142 ≈ 26(P√142)12 + 24

 

where,

            P√17 = (1+√17+√r1)/4;  r1 = 2(1+√17)

            P√73 = (5+√73+√r2)/4;  r2 = 2(41+5√73)

            P√97 = (9+√97+√r3)/4;  r3 = 2(81+9√97)

            P√193 = (13+√193+√r4)/2;  r4 = 2(179+13√193)

 

            P√14 = (1+√2+√r5)/2;  r5 = (-1+2√2)

            P√34 = (3+√17+√r6)/4;  r6 = 2(5+3√17)

            P√46 = (3+√2+√r7)/2;  r7 = (7+6√2)

            P√82 = (9+√41+√r8)/4;  r8 = 2(53+9√41)

            P√142 = (9+5√2+√r9)/2;  r9 = (127+90√2)

 

Others:

           

            eπ√70 ≈ 26(P√70)12 + 24                        eπ√85 ≈ 26(P√85)6 - 24

            eπ√130 ≈ 26(P√130)12 + 24                     eπ√133 ≈ 26(P√133)6 - 24

            eπ√190 ≈ 26(P√190)12 + 24                     eπ√253 ≈ 26(P√253)6 - 24

                                                                       

            eπ√30 ≈ 26(P√30)4 + 24                          eπ√33 ≈ 26(P√33)4 - 24

            eπ√42 ≈ 26(P√42)4 + 24                          eπ√57 ≈ 26(P√57)4 - 24

            eπ√78 ≈ 26(P√78)4 + 24                          eπ√177 ≈ 26(P√177)4 - 24

            eπ√102 ≈ 26(P√102)4 + 24

                                                                        eπ√21 ≈ 26(P√21)2 - 24

eπ√93 ≈ 26(P√93)2 - 24 

 

where, 

            P√70 = (f)2 (1+√2)

            P√130 = (f)3 (3+√13)/2

            P√190 = (f)3 (3+√10)

 

            P√30 = (f)3 (3+√10) (Curious, same as above.)

            P√42 = (7+2√14) (14+3√21)/7

            P√78 = (3+√13)3 (5+√26)/8

            P√102 = (2+√2)3 (3√2+√17)2/√8

 

            P√85 = (f)4 (9+√85)/2

            P√133 = (3+√7)2 (5√7+3√19)/4

            P√253 = (5+√23)2 (13√11+9√23)/4

            P√33 = (1+√3)3 (3+√11)/4

            P√57 = (1+√3)3 (13+3√19)/4

            P√177 = (1+√3)9 (23+3√59)/32

            P√21 = (√3+√7)3 (3+√7)2/16

            P√93 = (3√3+√31)3 (39+7√31)2/16

 

            For the next class numbers 6, 8, 10, 12, while I have the complete list for groups 1 and 2, we will give only one example per group to illustrate a certain pattern.

 

Class Number 6

 

Let, y = x – 1/x

 

Group 1: eπ√29 ≈ 26x6 – 24; (y3-9y2+8y-16=0)

 

Group 2: eπ√26 ≈ 26x12 + 24; (y3-2y2+y-4=0)

 

Class Number 8

 

Let, y = x + 1/x

 

Group 1: eπ√41 ≈ 26x12 – 24; (y4-5y3+3y2+3y+2=0)

 

Group 2: eπ√62 ≈ 26x12 + 24; (y4-2y3-17y2-24y-8=0)

 

Class Number 10

 

Let, y = x – 1/x

 

Group 1: eπ√181 ≈ 26x6 – 24; (y5-573y4-81y3-3483y2-3240y-3888=0)

 

Group 2: eπ√74 ≈ 26x12 + 24; (y5-8y4+14y3-36y2+41y-28=0)

 

Class Number 12

 

Let, y = x + 1/x

 

Group 1: eπ√89 ≈ 26x12 – 24; (y6-5y5-27y4-25y3+28y2+44y+16=0)

 

Group 2: eπ√274 ≈ 26x12 + 24; (y6-57y5+168y4-78y3+45y2-345y+202=0)

 

and so on…

 

We can summarize our results.  Given even discriminants 4p or 8q of even class numbers with p or q prime.  For class numbers 2, 6, 10,… (4m+2),  let, y = x – 1/x, where y is the appropriate root of an equation of degree (4m+2)/2 = 2m+1:

 

Group 1: eπ√p ≈ 26x6 – 24;         Group 2: eπ√(2q) ≈ 26x12 + 24

 

For class numbers 4, 8, 12,… (4m+4),  let, y = x + 1/x, where y is the appropriate root of an equation of degree (4m+4)/2 = 2m+2:

 

Group 1: eπ√p ≈ 26x12 – 24;        Group 2: eπ√(2q) ≈ 26x12 + 24

 

In other words, for what we defined as groups 1 and 2, we can observe two things: (a) Let x be the appropriate root of the Weber class polynomial.  For group 1, eπ√p is closely approximated by 26x6 for class number 4m+2 but 26x12 for class number 4m+4.  For group 2, there is no difference and eπ√(2q) is closely approximated by 26x12.  (b) For class number 4m+2, the Weber class polynomial in x is a semi-palindromic polynomial, the same whether read forwards or backwards but only if we disregard sign.  However, for class number 4m+4, it is a true palindromic polynomial.

 

IV. Ramanujan Class Polynomials

 

            The modular function involved in the last pair of examples has a formal designation in another context, the Monster group, which we will be going into later.  However, for purposes of consistency, perhaps it is permissible to call it as the r-function (for Ramanujan) since he did work on this function.  This function, r(q), has the series expansion,

 

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

 

and we see why the last pair of examples involved approximations to 104.  Just like j(q) and w(q), r(q) again is an algebraic number determined by an equation of degree k dependent on the class number n of some discriminant d.  This equation, which perhaps we can call the Ramanujan class polynomial, is solvable in radicals.

            The r(q) given below for class numbers 2 and 4 were known to Ramanujan, though for d = 14, 82, 42, 190, it doesn’t seem to be found in his Notebooks. 

 

Class Number 2

 

eπ√5 ≈ (4Ö2)4 + 100

            eπ√13 ≈ (12Ö2)4 + 104

            eπ√37 ≈ (84Ö2)4 + 104

 

            eπ√6 ≈ (4Ö3)4 - 106

            eπ√10 ≈ 124 -104

            eπ√22 ≈ (12Ö11)4 -104

            eπ√58 ≈ 3964 -104

 

Class Number 4

 

(Unknown for eπ√17, eπ√73, eπ√97, eπ√193.)

 

            eπ√14 ≈ 44(11+8√2)2 - 104

            eπ√34 ≈ 124(4+√17)4 - 104

            eπ√46 ≈ 124(147+104√2)2 - 104

            eπ√82 ≈ 124(51+8√41)4 - 104

            eπ√142 ≈ 124(467539+330600√2)2 - 104

 

            eπ√30 ≈ (4√3)4(5+4√2)4 - 104

            eπ√42 ≈ 44(21+8√6)4 - 104

            eπ√78 ≈ (4√3)4(75+52√2)4 - 104

            eπ√102 ≈ (4√3)4(200+49√17)4 - 104

 

            eπ√70 ≈ (12√7)4(5√5+8√2)4 - 104

            eπ√130 ≈ 124(323+40√65)4 - 104

            eπ√190 ≈ (12√19)4(481+340√2)4 - 104

 

Class Number 6

 

The r(q) for class number 6 was found by myself, using an assumption and again the Integer Relations applet.  I observed that, in the r(q) for class number 2 for what we defined as group 2 (namely d = 6, 10, 22, 58) for d = 6 & 22, r(q) was a quadratic irrational, while for d = 10 & 58, r(q) was an integer.  The difference was that d of the former was of the form 2(4m-1) while for the latter was 2(4m+1).  Since we already know that the degree k of the r(q) can be dependent on the nature of d, might it be the case that for d = 2(4m+1) of class number n, then eπ√d ≈ x4 – 104, where x is a root of an equation of degree n/2?

It seems it was the case.  A check to the validity of the four cubics below can be made considering the polynomial discriminants are given by 3d.  It is hoped that an interested reader can provide the missing polynomials for class number 8 and above.

 

            eπ√26 ≈ (4x)4 – 104; (x3-13x2-9x-11=0)

            eπ√106 ≈ (12x)4 – 104; (x3-271x2+63x-49=0)

            eπ√202 ≈ (12x)4 – 104; (x3-5871x2+2815x-913=0)

            eπ√298 ≈ (12x)4 – 104; (x3-64419x2-16061x-1441=0)

 

Class Number 8

 

            eπ√178 ≈ (12x)4 – 104; x = ?

            eπ√226 ≈ (12x)4 – 104; x = ?

            eπ√466 ≈ (12x)4 – 104; x = ?

            eπ√562 ≈ (12x)4 – 104; x = ?

 

            I am aware of r(q) only for class numbers 2, 4, 6 so far, or only for even n.  Ramanujan nor others does not seem to have worked on class polynomials defining r(q) for odd n.  It should be interesting to know if indeed there are such polynomials.

 

V.  Pi Formulas

 

We can use our modular functions j(q), r(q), and perhaps also w(q) to come up with formulas for pi, or more accurately 1/p.  We have the following infinite series due to the Chudnovsky brothers (where the summation S is understood to go from n = 0 to infinity),

 

Let, c = (-1)n(6n)!/((n!)3(3n)!)

 

            1/(4p) = S c (154n+15)/(323)n+1/2

            1/(12p) = S c (342n+25)/(963)n+1/2

            1/(12p) = S c (16254n+789)/(9603)n+1/2

            1/(12p) = S c (261702n+10177)/(52803)n+1/2

            1/(12p) = S c (545140134n+13591409)/(6403203)n+1/2

 

which uses the j(q) of d = 11, 19, 43, 67, 163 of class number 1.  The “signature” of the d can be found in the formula, other than the j(q) in the denominator.  Consider the factorizations,

 

            154 = 2*7*11

            342 = 2*32 *19

            16254 = 2*33 *7*43

            261702 = 2*32 *7*31*67

            545140134 = 2*32 *7*11*19*127*163

 

The general form of the formula seems to be:

 

            1/(12p) = S c (An+B)/(C)n+1/2

 

where A, B, C are algebraic numbers of degree k.  Thus, one can also use the j(q) of the d of class number 2 and so on. 

            The inspiration for the formulas derived by the Chudnovskys was a set of beautiful formulas for 1/p (17 in all) found by Ramanujan and listed down in his notebooks with little explanation on how he came up with them.  Most of them involve d of class number 2.  What I’m interested are the two formulas:

 

            1/(pÖ8) = 1/32 S r (10n+1)/124n

            1/(pÖ8) = 1/992 S r (26390n+1103)/3964n

 

where, r = (4n)!/(n!4).  To recall,

 

            eπ√10 ≈ 124 – 104

            eπ√58 ≈ 3964 – 104

 

The two formulas use the r(q) of the above.  (Note that 10 = 2*5 and 26390 = 2*5*7*13*29.)  I believe the general form is,

 

            1/(pÖ8) = 1/D S r (An+B)/C4n

 

where A, B, C, D are algebraic integers of degree k and C4 is the r(q) of a d of class number 2k.  Thus, it will be restricted to even class numbers.  It may then be a slightly different general form to the one found by the Borweins, though I’m not sure if they will turn out to be essentially the same.  The next candidates will be d = 34, 82 with class number 4 and r(q) of algebraic degree 2,

 

            eπ√34 ≈ 124(4+√17)4 – 104

            eπ√82 ≈ 124(51+8√41)4 – 104

 

and which should have A, B, D also as algebraic numbers of degree 2 if my assumption is correct.  I am not aware of pi formulas that use the Weber function w(q) though I believe one can perhaps find general forms in analogy with what was done for j(q) and r(q).

 

VI.  Monster Group and Conclusion

 

            Before we go to a fascinating connection to group theory and conclude our paper, we can make a small clarification regarding the constant epÖ163.  Ramanujan worked mostly on d with class number a power of two, and while epÖ58 is found in his notebooks, epÖ163 is not.  Hermite was aware of it as being an almost-integer c. 1859. 

The name is taken from an April Fool’s joke by Martin Gardner where he claimed Ramanujan had conjectured that it was exactly an integer.  In fact, numbers of the form epÖd for positive integer d are transcendental, as proven by Aleksandr Gelfond.  However, this interesting property of  epÖ163 seems to be in line with the body of Ramanujan’s work which itself is most interesting, especially keeping in mind the conditions in which it was made.  So it turns out the name for this constant is fitting indeed.

We pointed out earlier that there is a connection between the modular functions we have mentioned and what is called the Monster group.  To recall, the series expansion of the j-function was,

 

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

 

Now, the Monster group M, the largest of the 26 sporadic groups, is the group of rotations in 196883-dimensional space.  Its irreducible representations are given by 1, 196883, 21296876…etc.  It was noticed by John McKay in the late 70’s that 196883 was awfully close to the coefficient 196884 of the j-function above. When John Conway was told by J.G. Thompson about this observation, he thought that it was “moonshine”, or fanciful.  However, when you realize that,

 

196884 = 1 + 196883

21493760 = 1 + 196883 + 21296876

 

and so on, or the coefficients of the j-function seemed to be simple linear combinations of the representations of the Monster, then something really interesting must be going on.  The assumed relationship between the j-function and the Monster was known as the Monstrous Moonshine Conjecture, after a paper written by Conway and S. Norton in 1979 and was finally proven to be true by Richard Borcherds in 1992.  And as if that amazing relationship was not enough, the proof used a theorem from string theory!  Borcherds eventually won the Fields medal for proving this conjecture.

            Thus, the coefficients of the j-function are also known as the McKay-Thompson series of class 1A for Monster.  And what about our other modular function w(q)?

 

            w(q) = 1/q + 24 + 276q + 2048q2 + 11202q3 + …

 

The coefficients of the w-function happen to be also connected to the Monster and is known as the McKay-Thompson series of class 2B for Monster.  For r(q)?

 

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

 

The list of coefficients is also known as the McKay-Thompson series of class 2A for Monster.

            And so we have this profound connection between two seemingly different mathematical topics.  Ramanujan would have loved this.

 

 

--End--

 

ã 2005

Titus Piezas III

January 14, 2005

http://www.oocities.com/titus_piezas/  ¬ (Click here for an index of papers)

tpiezas@uap.edu.ph

 

 

 

References

 

  1. Francois Morain, “Construction of Hilbert Class Fields of Imaginary Quadratic Fields and Dihedral Equations Modulo p”
  2. Srinivasa Ramanujan and Bruce Berndt, Ramanujan’s Notebooks, Springer-Verlag, 2nd Ed., 1989
  3. N.J. Sloane, Online Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/
  4. Heinrich Weber, Lehrbuch der Algebra, Chelsea, 1961
  5. Annegret Weng, Class Polynomials of CM-Fields, http://www.exp-math.uni-essen.de/zahlentheorie/classpol/class.html
  6. Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chapman & Hall, 1999 or http://mathworld.wolfram.com/
  7. Noriko Yui and Don Zagier, “On The Singular Values of Weber Modular Functions”, Mathematics of Computation, Vol. 66, Number 220, Oct 1997
  8. et al.
1