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Dedekind eta function

For the Dirichlet series, see Dirichlet eta function.
File:Dedekind Eta.jpg
Dedekind η-function in the complex plane

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.

Definition

For any such complex number τ, let q = exp(2πiτ), and define the eta function by,

<math>\eta(\tau) = e^{\frac{\pi \rm{i} \tau}{12}} \prod_{n=1}^{\infty} (1-q^{n}) .</math>

The notation <math>q \equiv e^{2\pi \rm{i} \tau}\,</math> is now standard in number theory, though many older books use q for the nome <math>e^{\pi \rm{i} \tau}\,</math>. Its 24th power gives,

<math>\Delta=(2\pi)^{12}\eta^{24}(\tau)</math>

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.

File:Q-Eulero.jpeg
Modulus of Euler phi on the unit disc, colored so that black=0, red=4
File:Discriminant real part.jpeg
The real part of the modular discriminant as a function of q.

The eta function satisfies the functional equations[1]

<math>\eta(\tau+1) =e^{\frac{\pi {\rm{i}}}{12}}\eta(\tau),\,</math>
<math>\eta(-\tfrac{1}{\tau}) = \sqrt{-{\rm{i}}\tau} \eta(\tau).\,</math>

More generally, suppose abcd are integers with ad − bc = 1, so that

<math>\tau\mapsto\frac{a\tau+b}{c\tau+d}</math>

is a transformation belonging to the modular group. We may assume that either c > 0, or c = 0 and d = 1. Then

<math>\eta \left( \frac{a\tau+b}{c\tau+d} \right) =

\epsilon (a,b,c,d) (c\tau+d)^{\frac{1}{2}} \eta(\tau),</math>

where

<math>\epsilon (a,b,c,d)=e^{\frac{b{\rm{i}} \pi}{12}}\quad(c=0,d=1);</math>
<math>\epsilon (a,b,c,d)=e^{{\rm{i}}\pi [\frac{a+d}{12c} - s(d,c)

-\frac{1}{4}]}\quad(c>0).</math>

Here <math>s(h,k)\,</math> is the Dedekind sum

<math>s(h,k)=\sum_{n=1}^{k-1} \frac{n}{k}

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

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

<math>\Delta(\tau) = (2 \pi)^{12} \eta(\tau)^{24}\,</math>

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:

<math>\eta(\tau) = \sum_{n=1}^\infty \chi(n) \exp(\tfrac{1}{12} \pi i n^2 \tau),</math>

where <math>\chi(n)</math> is the Dirichlet character modulo 12 with <math>\chi(\pm1) = 1</math>, <math>\chi(\pm 5)=-1</math>. Explicitly,

<math>\eta(\tau) = e^{\tfrac{\pi i \tau}{12}}\vartheta_3(\tfrac{\pi(\tau+1)}{2}, e^{3\pi i \tau}).</math>

The Euler function

<math>\phi(q) = \prod_{n=1}^{\infty} \left(1-q^n\right),</math>

related to <math>\eta \,</math> by <math>\phi(q)= q^{-1/24} \eta(\tau)\,</math>, has a power series by the Euler identity:

<math>\phi(q)=\sum_{n=-\infty}^\infty (-1)^n q^{(3n^2-n)/2}.</math>

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 <math>q^{1/24}</math> 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.

Special values

The above connection with the Euler function together with the special values of the latter, it can be easily deduced that

<math>

\eta(i)=\frac{\Gamma \left(\frac{1}{4}\right)}{2 \pi ^{3/4}}, </math>

<math>

\eta\left(\tfrac{1}{2}i\right)=\frac{\Gamma \left(\frac{1}{4}\right)}{2^{7/8} \pi ^{3/4}}, </math>

<math>

\eta(2i)=\frac{\Gamma \left(\frac{1}{4}\right)}{2^{{11}/8} \pi ^{3/4}}, </math>

<math>

\eta(4i)=\frac{\sqrt[4]{-1+\sqrt{2}}\; \Gamma \left(\frac{1}{4}\right)}{2^{{29}/16} \pi ^{3/4}}. </math>

Eta quotients

Quotients of the Dedekind eta function at imaginary quadratic arguments may be algebraic, while combinations of eta quotients may even be integral. For example, define,

<math>j(\tau)=\Big(\big(\tfrac{\eta(\tau)}{\eta(2\tau)}\big)^{8}+2^8 \big(\tfrac{\eta(2\tau)}{\eta(\tau)}\big)^{16}\Big)^3</math>
<math>j_{2A}(\tau)=\Big(\big(\tfrac{\eta(\tau)}{\eta(2\tau)}\big)^{12}+2^6 \big(\tfrac{\eta(2\tau)}{\eta(\tau)}\big)^{12}\Big)^2</math>
<math>j_{3A}(\tau) =\Big(\big(\tfrac{\eta(\tau)}{\eta(3\tau)}\big)^{6}+3^3 \big(\tfrac{\eta(3\tau)}{\eta(\tau)}\big)^{6}\Big)^2</math>

then,

<math>j\Big(\tfrac{1+\sqrt{-163}}{2}\Big) = -640320^3,\quad e^{\pi\sqrt{163}} \approx 640320^3+743.99999999999925\dots</math>
<math>j_{2A}\Big(\tfrac{\sqrt{-58}}{2}\Big) = 396^4,\qquad \quad e^{\pi\sqrt{58}}\approx 396^4-104.00000017\dots</math>
<math>j_{3A}\Big(\tfrac{1+\sqrt{-89/3}}{2}\Big) = -300^3,\quad e^{\pi\sqrt{89/3}}\approx 300^3+41.999971\dots</math>

and so on, values which appear in Ramanujan–Sato series.

See also

References

  1. ^ Siegel, C.L. "A Simple Proof of <math>\eta(-1/\tau) = \eta(\tau)\sqrt{\tau/{\rm{i". }\,</math>|journal=Mathematika|year=1954|volume=1|page=4|doi=10.1112/S0025579300000462}}
  • Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (2 ed), Graduate Texts in Mathematics 41 (1990), Springer-Verlag, ISBN 3-540-97127-0 See chapter 3.
  • Neal Koblitz, Introduction to Elliptic Curves and Modular Forms (2 ed), Graduate Texts in Mathematics 97 (1993), Springer-Verlag, ISBN 3-540-97966-2