The Dedekind eta function, named after Richard Dedekind, is a function defined on the upper half-plane of complex numbers whose imaginary part is positive. For any such complex number $\backslash tau$, we define q = e^2iπτ, and define the eta function by

$\backslash eta(\backslash tau)\; =\; q^\{1/24\}\; \backslash prod\_\{n=1\}^\{\backslash infty\}\; (1-q^\{n\}).$

(The notation q = e^2iπτ is now standard in number theory, though many older books use q for the nome e^iπτ).

The eta function is holomorphic on the upper half-plane but cannot be continued analytically beyond it.

The eta function satisfies the functional equations

$\backslash eta(\backslash tau+1)\; =\; \backslash exp(2\; \backslash pi\; i/24)\backslash eta(\backslash tau),$

$\backslash eta(-1/\backslash tau)\; =\; \backslash sqrt\; \{-i\backslash tau\}\; \backslash eta(\backslash tau).$

More generally,

$\backslash eta\; \backslash left(\; \backslash frac\{a\backslash tau+b\}\{c\backslash tau+d\}\; \backslash right)\; =$

\epsilon (a,b,c,d) \left( -i(c\tau+d) \right)^{1/2} \eta(\tau)

where a, b, c, d are integers, with ad − bc = 1, and thus being a transform belonging to the modular group, and

$\backslash epsilon\; (a,b,c,d)=\backslash exp\; i\backslash pi\; \backslash left($

\frac{a+d}{12c} + s(-d,c) \right)

and s(h, k) is the Dedekind sum

$s(h,k)=\backslash sum\_\{n=1\}^\{k-1\}\; \backslash frac\{n\}\{k\}$

\left( \frac{hn}{k} - \left\lfloor \frac{hn}{k} \right\rfloor -\frac{1}{2} \right).

Because of these functional equations the eta function is a modular form of weight 1/2 and level 1 for a certain character or 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

$\backslash Delta(\backslash tau)\; =\; (2\; \backslash pi)^\{12\}\; \backslash eta(\backslash tau)^\{24\}$

and is a modular form of weight 12. (Some authors omit the factor of (2π)¹², so that the series expansion has integral coefficients). Because the eta function is easy to compute, it is often helpful 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

$\backslash phi(q)\; =\; \backslash prod\_\{n=1\}^\{\backslash infty\}\; \backslash left(1-q^n\backslash right)$

Note that the additional factor of $q^\{1/24\}$ between this and the Dedekind 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.

Note that the Jacobi triple product implies that the eta is (up to a factor) a Jacobi theta function for special values of the arguments.

See also

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• q-series
• Weierstrass's elliptic functions
• partition function (number theory)
• Kronecker limit formula

References

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• Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York. ISBN 0-387-97127-0 See chapter 3.

fractals | modular forms | elliptic functions

Dedekindsche η-Funktion | Fonction eta de Dedekind