j-Function

The -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 -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 -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 modular functions , , , , and as

(Weber 1979, p. 179; Atkin and Morain 1993).

The -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

is therefore related to via

The -function and its Fourier series satisfy the following properties:

1. for and ,

2. all are integers and

3. is an algebraic number, sometimes a rational number, and sometimes even an integer at values of in an imaginary quadratic field.

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

(OEIS A000521) are positive integers (Rankin 1977, Apostol 1997). Berwick (1916) calculated the first seven , Zuckerman (1939) found the first 24, and van Wijngaarden (1953) gave the first 100.

Some remarkable sum formulas involving for , where is the upper half-plane, and include

where is an Eisenstein series, is a q-Pochhammer symbol, and

where is the divisor function, and is the tau function (not to be confused with the half-period ratio ). In addition,

(Lehmer 1942; Apostol 1997, p. 92). These are closely related to Eisenstein series.

Equation (18) leads immediately to the remarkable congruence

Lehmer (1942) showed that

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

More generally, for and positive integer ,

(Lehner 1949ab; Apostol 1997, p. 91). There is no corresponding divisibility of every by 13. Newman (1958) showed

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

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

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

so

The imaginary quadratic field has fundamental discriminant when and when . Then is an algebraic integer whose minimal polynomial over has polynomial degree , where is the class number for fundamental discriminant (Silverman 1986; Berndt 1994, p. 90).

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

The positions of these special values of are illustrated above. (Note the curious though not particularly significant fact that number 5280 is also the number of feet in a mile.)

For odd Heegner numbers , the leading term in the Fourier series explains why is close to an integer. The best approximations with are

(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 with the greatest absolute fundamental discriminant among those of class number one), 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 negative fundamental discriminants having class number . The odd discriminants not divisible by three correspond to the exact values

and the even discriminants for , 10, 13, 22, 37, 58 correspond to

and discriminants divisible by 3,

with each squared factor the square of a fundamental unit.

The best approximations for are, for even discriminants,

and for odd discriminants,

The numbers

are also almost integers. These correspond to binary quadratic forms with discriminants , , and , 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).