page_title: Dedekind eta function
section_title: Definition
hierarchical_section_title: Dedekind eta function / Definition
caption_attribution_description: English: This picture shows the modulus on the complex plane, inside the unit circle , where is Euler's q-series for the Partition of an integer. The fractal self-similarity of this function is that of the modular group; note that this function is closely related to a modular form, the modular discriminant. Every modular function will have this general kind of self-similarity. It is colored with a rainbow of colors, with black representing values near zero, and red values near four. Note the Modular group symmetry, and the general resemblance to colorings of the interior of the Mandelbrot set. Italiano: L'immagine mostra il modulo nel piano complesso, all'interno del cerchio unitario , dove è la q-serie di Eulero per la partizione di un intero.
context_page_description: In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string theory.
context_section_description: For any complex number  with , let , then the eta function is defined by,
  The notation  is now standard in number theory, though many older books use q for  the nome . Raising the eta equation to the 24th power and multiplying by (2π)¹² gives
  where Δ is the modular discriminant.  The presence of 24 can be understood by connection with other occurrences, such as in the 24-dimensional Leech lattice.
  The eta function is holomorphic on the upper half-plane but cannot be continued analytically beyond it.
  The eta function satisfies the functional equations
  More generally, suppose a, b, c, d are integers with ad − bc = 1, so that
  is a transformation belonging to the modular group.  We may assume that either c > 0, or c = 0 and d = 1.  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  is "the" Dirichlet character modulo 12 with ,
  . Explicitly,
  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  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.
caption: Modulus of Euler phi on the unit disc, colored so that black=0, red=4
exif: {}