The Dedekind eta function, named after Richard Dedekind, is a function defined on the upper halfplane of complex numbers, where the imaginary part is positive. For any such complex number , we define , and define the eta function by
(The notation is now standard in number theory, though many older books use q for the nome ). The presence of 24 can be understood by connection with other occurrences, as in the modular discriminant and the Leech lattice.
The eta function is holomorphic on the upper halfplane but
cannot be continued analytically beyond it.
The eta function satisfies the functional equations
More generally, suppose are integers with , so that
is a transformation belonging to the modular group. We may assume that either , or and . Then
where
Here is the Dedekind sum
Because of these functional equations the eta function is a modular form of weight 1/2 and level 1 for a certain character of order 24 of the metaplectic double cover of the modular group, and can be used to define other modular forms. In particular the modular discriminant of Weierstrass can be defined as
and is a modular form of weight 12. (Some authors omit the factor of (2π)^{12}, so that the series expansion has integral coefficients).
The Jacobi triple product implies that the eta is (up to a factor) a Jacobi theta function for special values of the arguments:
where χ(n) is the Dirichlet character modulo 12 with , .
The Euler function
related to by , has a power series by the Euler identity:
Because the eta function is easy to compute numerically from either power series, it is often helpful in computation to express other functions in terms of it when possible, and products and quotients of eta functions, called eta quotients, can be used to express a great variety of modular forms.
The picture on this page shows the modulus of the Euler function: the additional factor of q^{1 / 24} between this and eta makes almost no visual difference whatsoever (it only introduces a tiny pinprick at the origin). Thus, this picture can be taken as a picture of eta as a function of q.
