**Ramanujan’s Constant (e ^{π√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,

e^{π√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 *e ^{π√d}* for some positive integer

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

e^{π√67} = 5280^{3}
+ 743.9999986…

e^{π√163} = 640320^{3}
+ 743.99999999999925…

including some relations that involve square roots,

e^{π√22
}≈ 2^{6}(1+√2)^{12} + 23.99988…

e^{π√58
}≈ 2^{6}((5+√29)/2)^{12} + 23.999999988…

and,

e^{π√42
}≈ 4^{4}(21+8√6)^{4} – 104.0000062…

e^{π√130
}≈ 12^{4}(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.

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 + 21493760q^{2} + 864299970q^{3} + …

where *q = e ^{2}*

The *j(q)* is dependent on the *t*. If *t* involves a quadratic irrational Ö*d*, 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 *e ^{π√d}*, if we know the class
number

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 *e ^{π√d}* involving integers, since

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 e^{π√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 e^{π√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)

e^{π√11
}≈ 32^{3 }+ 738

e^{π√19
}≈ 96^{3} + 744

e^{π√43
}≈ 960^{3} + 744

e^{π√67
}≈ 5280^{3} + 744

e^{π√163
}≈ 640320^{3} + 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,

e^{π√16
}≈ 66^{3 }- 744

e^{π√28
}≈ 255^{3} - 744

__Class Number 2__ (18
discriminants)

e^{π√20
}≈ 2^{3}(25+13√5)^{3} - 744

e^{π√52
}≈ 30^{3}(31+9√13)^{3} - 744

e^{π√148
}≈ 60^{3}(2837+468√37)^{3 }- 744

e^{π√24
}≈ 12^{3}(1+Ö2)^{2}(5+2Ö2)^{3} - 744

e^{π√40
}≈ 6^{3}(65+27√5)^{3 }- 744

e^{π√88
}≈ 60^{3}(155+108√2)^{3 }- 744

e^{π√232
}≈ 30^{3}(140989+26163√29)^{3 }- 744

e^{π√35
}≈ 16^{3}(15+7√5)^{3 }+ 744

e^{π√91
}≈ 48^{3}(227+63√13)^{3 }+ 744

e^{π√115
}≈ 48^{3}(785+351√5)^{3 }+ 744

e^{π√187
}≈ 240^{3}(3451+837√17)^{3 }+ 744

e^{π√235
}≈ 528^{3}(8875+3969√5)^{3 }+ 744

e^{π√403
}≈ 240^{3}(2809615+779247√13)^{3 }+ 744

e^{π√427
}≈ 5280^{3}(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 e^{π√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 e^{π√51
}as,

e^{π√51
}≈ 4*48^{3}(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:

e^{π√15
}≈ 3^{3}((1+Ö5)/2)^{2}(5+4Ö5)^{3 }+ 743

e^{π√51
}≈ 48^{3}(4+Ö17)^{2}(5+Ö17)^{3} + 744

e^{π√123
}≈ 480^{3}(32+5Ö41)^{2}(8+Ö41)^{3} + 744

e^{π√267
}≈ 240^{3}(500+53Ö89)^{2}(625+53Ö89)^{3} + 744 (1.0 x 10^{-17})

Note that:

(1/2)^{2
}- 5*(1/2)^{2 }= -1

4^{2
}-17*1^{2} = -1

32^{2}
- 41*5^{2} = -1

500^{2
}- 89*53^{2} = -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 e^{π√59}
and e^{π√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),

y^{3}+3491750y^{2}-5151296875y+23375^{3
}= 0

by setting y = x^{3}, 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 = x^{3},^{ }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,

e^{π√23
}≈ 5^{3}x^{3} + 744; (x^{3}-31x^{2}+26x-187=0)

e^{π√31
}≈ 3^{3}x^{3} + 744; (x^{3}-114x^{2}+93x-4301=0)

e^{π√59
}≈ 32^{3}x^{3} + 744; (x^{3}-98x^{2}+67x-22=0)

e^{π√83
}≈ 160^{3}x^{3} + 744; (x^{3}-87x^{2}+5x-2=0)

__Class Number 4__ (54
discriminants)

e^{π√55
}≈ x^{3 }+ 744; (x^{4}-2355x^{3}-8370x^{2}-5553900x-26484975=0)

e^{π√56
}≈ 4^{3}x^{3} – 744; (x^{4}-646x^{3}+8347x^{2}-11286x+84337=0)

__Class Number 5__ (25
discriminants)

e^{π√47
}≈ 5^{3}x^{3} + 744; (x^{5}-264x^{4}+484x^{3}-15419x^{2}+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 + 2048q^{2} +
11202q^{3} + …

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,

e^{π√d
}≈ x^{24} - 24

where *x* is a root of an equation of degree *3n* with a constant term *-2 ^{n}*. (The exception to this rule seems to be d =
3, which has a Weber class polynomial that is not a cubic.) If a discriminant

e^{π√d
}≈ 2^{12}x^{24} – 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 (e^{π√d}
+ 24)^{1/24} and (e^{π√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*)

__ __

e^{π√11
}≈ x^{24} – 24; (x^{3}-2x^{2}+2x-2=0)

e^{π√19
}≈ x^{24} – 24; (x^{3}-2x-2=0)

e^{π√43
}≈ x^{24} – 24; (x^{3}-2x^{2}-2=0)

e^{π√67
}≈ x^{24} – 24; (x^{3}-2x^{2}-2x-2=0)

e^{π√163
}≈ x^{24} – 24; (x^{3}-6x^{2}+4x-2=0)

__Class Number 3 __(16
discriminants, 2 of the form *8m+7*)

e^{π√23
}≈ 2^{12}x^{24} – 24; (x^{3}-x-1=0)

e^{π√31
}≈ 2^{12}x^{24} – 24; (x^{3}-x^{2}-1=0)

__ __

__Class Number 5 __(25
discriminants, 4 of the form *8m+7*)

e^{π√47
}≈ 2^{12}x^{24} – 24; (x^{5}-x^{3}-2x^{2}-2x-1=0)

e^{π√79
}≈ 2^{12}x^{24} – 24; (x^{5}-3x^{4}+2x^{3}-x^{2}+x-1=0)

e^{π√103
}≈ 2^{12}x^{24} – 24; (x^{5}-x^{4}-3x^{3}-3x^{2}-2x-1=0)

e^{π√127}≈
2^{12}x^{24} – 24; (x^{5}-3x^{4}-x^{3}+2x^{2}+x-1=0)

__Class Number 7 __(31
discriminants, 5 of the form *8m+7*)

e^{π√71
}≈ 2^{12}x^{24} – 24; (x^{7}-2x^{6}-x^{5}+x^{4}+x^{3}+x^{2}-x-1=0)

e^{π√151
}≈ 2^{12}x^{24} – 24; (x^{7}-3x^{6}-x^{5}-3x^{4}-x^{2}-x-1=0)

e^{π√223
}≈ 2^{12}x^{24} – 24; (x^{7}-5x^{6}-x^{4}-4x^{3}-x^{2}-1=0)

e^{π√463
}≈ 2^{12}x^{24} – 24; (x^{7}-11x^{6}-9x^{5}-8x^{4}-7x^{3}-7x^{2}-3x-1=0)

e^{π√487
}≈ 2^{12}x^{24} – 24; (x^{7}-13x^{6}+4x^{5}-4x^{4}+7x^{3}-4x^{2}+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 e^{π√p}
and e^{π√(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

e^{π√5
}≈ 2^{6}(*f*)^{6} – 24 e^{π√6
}≈ 2^{6}(1+√2)^{4 }+ 24

e^{π√13
}≈ 2^{6}((3+√13)/2)^{6} – 24 e^{π√10 }≈
2^{6}(*f*)^{12}
+ 24

e^{π√37
}≈ 2^{6}(6+√37)^{6} – 24 e^{π√22 }≈
2^{6}(1+√2)^{12} +24

e^{π√58
}≈ 2^{6}((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 e^{π√15}, are
almost-roots of sextics.)

__Class Number 4__

Group 1 Group 2

e^{π√17
}≈ 2^{6}(P√17)^{12} – 24 e^{π√14 }≈
2^{6}(P√14)^{12} + 24

e^{π√73
}≈ 2^{6}(P√73)^{12} – 24 e^{π√34 }≈
2^{6}(P√34)^{12} + 24

e^{π√97
}≈ 2^{6}(P√97)^{12} – 24 e^{π√46 }≈
2^{6}(P√46)^{12} + 24

e^{π√193
}≈ 2^{6}(P√193)^{12} – 24 e^{π√82 }≈
2^{6}(P√82)^{12} + 24

e^{π√142
}≈ 2^{6}(P√142)^{12} + 24

where,

P√17
= (1+√17+√r_{1})/4;
r_{1 }= 2(1+√17)

P√73
= (5+√73+√r_{2})/4;
r_{2 }= 2(41+5√73)

P√97
= (9+√97+√r_{3})/4;
r_{3} = 2(81+9√97)

P√193
= (13+√193+√r_{4})/2;
r_{4} = 2(179+13√193)

P√14
= (1+√2+√r_{5})/2;
r_{5} = (-1+2√2)

P√34
= (3+√17+√r_{6})/4;
r_{6} = 2(5+3√17)

P√46
= (3+√2+√r_{7})/2;
r_{7} = (7+6√2)

P√82
= (9+√41+√r_{8})/4;
r_{8} = 2(53+9√41)

P√142
= (9+5√2+√r_{9})/2;
r_{9} = (127+90√2)

*Others*:

e^{π√70
}≈ 2^{6}(P√70)^{12} + 24 e^{π√85 }≈
2^{6}(P√85)^{6} - 24

e^{π√130
}≈ 2^{6}(P√130)^{12} + 24 e^{π√133 }≈
2^{6}(P√133)^{6} - 24

e^{π√190
}≈ 2^{6}(P√190)^{12} + 24 e^{π√253 }≈
2^{6}(P√253)^{6} - 24

e^{π√30
}≈ 2^{6}(P√30)^{4} + 24 e^{π√33 }≈ 2^{6}(P√33)^{4}
- 24

e^{π√42
}≈ 2^{6}(P√42)^{4} + 24 e^{π√57 }≈ 2^{6}(P√57)^{4}
- 24

e^{π√78
}≈ 2^{6}(P√78)^{4} + 24 e^{π√177 }≈ 2^{6}(P√177)^{4}
- 24

e^{π√102
}≈ 2^{6}(P√102)^{4} + 24

e^{π√21
}≈ 2^{6}(P√21)^{2} - 24

e^{π√93
}≈ 2^{6}(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: e^{π√29 }≈
2^{6}x^{6} – 24; (y^{3}-9y^{2}+8y-16=0)

Group 2: e^{π√26 }≈
2^{6}x^{12} + 24; (y^{3}-2y^{2}+y-4=0)

__Class Number 8__

Let, y = x + 1/x

Group 1: e^{π√41 }≈
2^{6}x^{12} – 24; (y^{4}-5y^{3}+3y^{2}+3y+2=0)

Group 2: e^{π√62 }≈
2^{6}x^{12} + 24; (y^{4}-2y^{3}-17y^{2}-24y-8=0)

__Class Number 10__

Let, y = x – 1/x

Group 1: e^{π√181 }≈
2^{6}x^{6} – 24; (y^{5}-573y^{4}-81y^{3}-3483y^{2}-3240y-3888=0)

Group 2: e^{π√74 }≈
2^{6}x^{12} + 24; (y^{5}-8y^{4}+14y^{3}-36y^{2}+41y-28=0)

__Class Number 12__

Let, y = x + 1/x

Group 1: e^{π√89 }≈
2^{6}x^{12} – 24; (y^{6}-5y^{5}-27y^{4}-25y^{3}+28y^{2}+44y+16=0)

Group 2: e^{π√274 }≈
2^{6}x^{12} + 24; (y^{6}-57y^{5}+168y^{4}-78y^{3}+45y^{2}-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: e^{π√p }≈
2^{6}x^{6} – 24; Group
2: e^{π√(2q) }≈ 2^{6}x^{12} + 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: e^{π√p }≈
2^{6}x^{12} – 24; Group
2: e^{π√(2q) }≈ 2^{6}x^{12} + 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, e^{π√p} is
closely approximated by *2 ^{6}x^{6}* for class number

**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 + 96256q^{2}
+ 1240002q^{3} + …

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.

e^{π√5
}≈ (4Ö2)^{4
}+ 100

e^{π√13
}≈ (12Ö2)^{4}
+ 104

e^{π√37
}≈ (84Ö2)^{4}
+ 104

e^{π√6
}≈ (4Ö3)^{4}
- 106

e^{π√10
}≈ 12^{4} -104

e^{π√22
}≈ (12Ö11)^{4}
-104

e^{π√58
}≈ 396^{4} -104

__Class Number 4__

(Unknown for e^{π√17},
e^{π√73}, e^{π√97}, e^{π√193}.)

e^{π√14
}≈ 4^{4}(11+8√2)^{2} - 104

e^{π√34
}≈ 12^{4}(4+√17)^{4} - 104

e^{π√46
}≈ 12^{4}(147+104√2)^{2} - 104

e^{π√82
}≈ 12^{4}(51+8√41)^{4} - 104

e^{π√142
}≈ 12^{4}(467539+330600√2)^{2} - 104

e^{π√30
}≈ (4√3)^{4}(5+4√2)^{4} - 104

e^{π√42
}≈ 4^{4}(21+8√6)^{4} - 104

e^{π√78
}≈ (4√3)^{4}(75+52√2)^{4} - 104

e^{π√102
}≈ (4√3)^{4}(200+49√17)^{4} - 104

e^{π√70
}≈ (12√7)^{4}(5√5+8√2)^{4} - 104

e^{π√130
}≈ 12^{4}(323+40√65)^{4} - 104

e^{π√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 e^{π√d
}≈ x^{4} – 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.

e^{π√26
}≈ (4x)^{4} – 104; (x^{3}-13x^{2}-9x-11=0)

e^{π√106
}≈ (12x)^{4} – 104; (x^{3}-271x^{2}+63x-49=0)

e^{π√202
}≈ (12x)^{4} – 104; (x^{3}-5871x^{2}+2815x-913=0)

e^{π√298
}≈ (12x)^{4} – 104; (x^{3}-64419x^{2}-16061x-1441=0)

e^{π√178
}≈ (12x)^{4} – 104; x = ?

e^{π√226
}≈ (12x)^{4} – 104; x = ?

e^{π√466
}≈ (12x)^{4} – 104; x = ?

e^{π√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.

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)/(32^{3})^{n+1/2}

1/(12p) = S c
(342n+25)/(96^{3})^{n+1/2}

1/(12p) = S c
(16254n+789)/(960^{3})^{n+1/2}

1/(12p) = S c
(261702n+10177)/(5280^{3})^{n+1/2}

1/(12p) = S c
(545140134n+13591409)/(640320^{3})^{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*3^{2 }*19

16254
= 2*3^{3 }*7*43

261702
= 2*3^{2 }*7*31*67

545140134
= 2*3^{2 }*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) =
1/3^{2} S
r (10n+1)/12^{4n}

1/(pÖ8) =
1/99^{2} S
r (26390n+1103)/396^{4n}

where, r = (4n)!/(n!^{4}). To recall,

e^{π√10
}≈ 12^{4} – 104

e^{π√58
}≈ 396^{4} – 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)/C^{4n}

where A, B, C, D are algebraic
integers of degree *k* and C^{4} 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,

e^{π√34
}≈ 12^{4}(4+√17)^{4} – 104

e^{π√82
}≈ 12^{4}(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)*.

Before
we go to a fascinating connection to group theory and conclude our paper, we
can make a small clarification regarding the constant e^{p}^{Ö163}. Ramanujan worked mostly on *d* with
class number a power of two, and while e^{p}^{Ö58 }is
found in his notebooks, e^{p}^{Ö163 }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 e^{p}^{Öd }for
positive integer *d* are transcendental, as proven by Aleksandr
Gelfond. However, this interesting
property of e^{p}^{Ö163 }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 + 21493760q^{2}
+ 864299970q^{3} + …

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 + 2048q^{2} +
11202q^{3} + …

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 + 96256q^{2}
+ 1240002q^{3} + …

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

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