The j-function is the modular function defined by

where is the half-period ratio, ,

is Klein's absolute invariant, is the elliptic lambda function

are Jacobi theta functions,

is the nome, and .

Gauss was apparently aware of the j-function before 1800. Hermite used it in solving the quintic in about 1858. Dedekind gave a nice definition in about 1877, and Klein studied the function beginning in 1879 or 1880. The j-function is related to the factors of the group order of the monster group and to supersingular primes (Ogg 1980).

This function can also be specified in terms of the Weber functions f, , , , and as

(Weber 1902, p. 179; Atkin and Morain 1993).

The j-function is an analytic function on the upper half-plane which is invariant with respect to the special linear group . It has a Fourier series

where

The coefficients in the expansion of the j-function satisfy:

1. for n < -1 and ,

2. all s are integers with fairly limited growth with respect to n, and

3. is an algebraic number, sometimes a rational number, and sometimes even an integer at certain very special values of .

The latter result is the end result of the massive and beautiful theory of complex multiplication and the first step of Kronecker's so-called "Jugendtraum."

Therefore all of the coefficients in the Laurent series

	(12)

(Sloane's A000521) are positive integers (Rankin 1977, Apostol 1997). Berwick (1916) calculated the first seven c(n), Zuckerman (1939) found the first 24, and van Wijngaarden (1963) gave the first 100.

Some remarkable sum formulas involving for , where H is the upper half-plane, and c(n) include

and

where is the divisor function and . In addition,

where is the tau function (Lehmer 1942; Apostol 1997, p. 92), not to be confused with the half-period ratio . These are closely related to Eisenstein series.

Equation (16) leads immediately to the remarkable congruence

Lehmer (1942) showed that

for all , and Lehner (1949) and Apostol (1997, pp. 22, 74, and 90-91) demonstrated that

	(19)
	(20)
	(21)
	(22)
	(23)

More generally,

	(24)
	(25)
	(26)
	(27)

(Lehner 1949; Apostol 1997, p. 91). Congruences of this type cannot exist for 13, but Newman (1958) showed

where and if x is not an integer (Apostol 1997, p. 91). Congruences for have been generalized by Atkin and O'Brien (1967).

An asymptotic formula for c(n) was discovered by Petersson (1932), and subsequently independently rediscovered by Rademacher (1938):

Let d be a squarefree positive integer, and define the half-period ratio by

so

It then turns out that is an algebraic integer of degree , where is the class number of the binary quadratic form discriminant of the quadratic field (Silverman 1986; Berndt 1994, p. 90).

If , then is a algebraic integer of degree 1, i.e., just a plain integer. Furthermore, the integer is a perfect cube. But these are precisely the Heegner numbers -1, -2, -3, -7, -11, -19, -43, -67, -163. The exact values of corresponding to the Heegner numbers are

The positions of these special values of are illustrated above. The number 5280 is particularly interesting since it is also the number of feet in a mile.

The greater (in absolute value) the Heegner number d, the closer to an integer is the expression , since the initial term in is the largest and subsequent terms are the smallest. The best approximations with are therefore

(the latter of which appears in Trott 2004, p. 8). The almost integer generated by the last of these, (corresponding to the field and the imaginary quadratic field of maximal discriminant), is sometimes known as the Ramanujan constant. However, this attribution is historically fallacious since this amazing property of was first noted by Hermite (1859) and does not seem to appear in any of the works of Ramanujan.

There are 18 numbers having class number , corresponding to the exact values

and so on.

The numbers

are also almost integers. These correspond to binary quadratic forms with discriminants -88, -148, and -232, which are the largest (in absolute value) discriminants with class number two that are divisible by 4. They were noted by Ramanujan (Berndt 1994, pp. 88-91).

 

Almost Integer, Heegner Number, Imaginary Quadratic Field, Klein's Absolute Invariant, Monster Group, Ramanujan Constant, Supersingular Prime, Weber Functions