Dedekind eta function

File:Dedekind Eta.jpg

Dedekind Template:Mvar-function in the upper half-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. It also occurs in bosonic string theory.

Definition

For any complex number Template:Mvar with Template:Math, let Template:Math; then the eta function is defined by,

<math>\eta(\tau) = e^\frac{\pi i \tau}{12} \prod_{n=1}^\infty \left(1-e^{2 n\pi i \tau}\right) = q^\frac{1}{24} \prod_{n=1}^\infty \left(1 - q^n\right) .</math>

Raising the eta equation to the 24th power and multiplying by Template:Math gives

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

where Template:Math 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 Template:Mvar.

The eta function satisfies the functional equations<ref>Template:Cite journal</ref>

<math>\begin{align}

\eta(\tau+1) &=e^\frac{\pi i}{12}\eta(\tau),\\ \eta\left(-\frac{1}{\tau}\right) &= \sqrt{-i\tau}\, \eta(\tau).\, \end{align}</math>

In the second equation the branch of the square root is chosen such that Template:Math when Template:Math.

More generally, suppose Template:Math are integers with Template:Math, 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 Template:Math, or Template:Math and Template:Math. Then

<math>\eta \left( \frac{a\tau+b}{c\tau+d} \right) = \epsilon (a,b,c,d) \left(c\tau+d\right)^\frac12 \eta(\tau),</math>

where

<math>\epsilon (a,b,c,d)= \begin{cases}

e^\frac{bi \pi}{12} &c=0,\,d=1, \\ e^{i\pi \left(\frac{a+d}{12c} - s(d,c)-\frac14\right)} &c>0. \end{cases}</math>

Here Template: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 -\frac12 \right).</math>

Because of these functional equations the eta function is a modular form of weight Template:Sfrac 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 the Weierstrass elliptic function with

<math>\omega_2=\tau\omega_1</math>

can be defined as

<math>\Delta(\tau) = (2 \pi\omega_1)^{12} \eta(\tau)^{24}\,</math>

and is a modular form of weight 12. Some authors omit the factor of Template:Math, 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:<ref>Template:Citation</ref>

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

where Template:Math is "the" Dirichlet character modulo 12 with Template:Math and Template:Math. Explicitly,Template:Citation needed

<math>\eta(\tau) = e^\frac{\pi i \tau}{12}\vartheta\left(\frac{\tau+1}{2}; 3\tau\right).</math>

The Euler function

<math>\begin{align}

\phi(q) &= \prod_{n=1}^\infty \left(1-q^n\right) \\ &= q^{-\frac{1}{24}} \eta(\tau), \end{align}</math>

has a power series by the Euler identity:

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

Note that by using Euler Pentagonal number theorem for <math> \mathfrak{I} (\tau )>0 </math>, the eta function can be expressed as

<math>\eta(\tau)=\sum_{n=-\infty}^\infty e^{\pi i n}e^{3\pi i \left(n+\frac{1}{6}\right)^2 \tau}.</math>

This can be proved by using <math>x=2\pi i \tau</math> in Euler Pentagonal number theorem with the definition of eta function.

Another way to see the Eta function is through the following limit

<math>\lim_{z \to 0} \frac{\vartheta_1(z|\tau)}{z}=2\pi \eta^3(\tau)</math>

Which alternatively is:

<math> \sum_{n=0}^\infty (-1)^n (2n+1)q^{\frac{(2n+1)^2}8}=\eta^3(\tau)</math>

Where <math> \vartheta_1(z|\tau)</math> is the Jacobi Theta function and <math> \vartheta_1(z|\tau)=-\vartheta_{11}(z;\tau)</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 Template:Math between this and eta makes almost no visual difference whatsoever. Thus, this picture can be taken as a picture of eta as a function of Template:Mvar.

Combinatorial identities

The theory of the algebraic characters of the affine Lie algebras gives rise to a large class of previously unknown identities for the eta function. These identities follow from the Weyl–Kac character formula, and more specifically from the so-called "denominator identities". The characters themselves allow the construction of generalizations of the Jacobi theta function which transform under the modular group; this is what leads to the identities. An example of one such new identity<ref>Template:Citation</ref> is

<math>\eta(8\tau)\eta(16\tau) = \sum_{m,n\in \mathbb{Z} \atop m \le |3n|}

(-1)^m q^{(2m+1)^2 - 32n^2} </math> where Template:Math is the [[q-analog|Template:Mvar-analog]] or "deformation" of the highest weight of a module.

Special values

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

<math>\begin{align}

\eta(i)&=\frac{\Gamma \left(\frac14\right)}{2 \pi ^\frac34} \\[6pt]

\eta\left(\tfrac{1}{2}i\right)&=\frac{\Gamma \left(\frac14\right)}{2^\frac78 \pi ^\frac34} \\[6pt]

\eta(2i)&=\frac{\Gamma \left(\frac14\right)}{2^\frac{11}{8} \pi ^\frac34} \\[6pt]

\eta(3i)&=\frac{\Gamma \left(\frac14\right)}{2\sqrt[3]{3} \left(3+2 \sqrt{3}\right)^\frac{1}{12} \pi ^\frac34} \\[6pt]

\eta(4i)&=\frac{\sqrt[4]{-1+\sqrt{2}}\, \Gamma \left(\frac14\right)}{2^\frac{29}{16} \pi ^\frac34} \\[6pt]

\eta\left(e^\frac{2 \pi i}{3}\right)&=e^{-\frac{\pi i}{24}} \frac{\sqrt[8]{3} \, \Gamma \left(\frac13\right)^\frac32}{2 \pi } \end{align}</math>

Eta quotients

Eta quotients are defined by quotients of the form

<math> \prod_{0<d\mid N}\eta(d\tau)^{r_d} </math>

where Template:Mvar is a non-negative integer and Template:Mvar is any integer. Linear combinations of eta quotients at imaginary quadratic arguments may be algebraic, while combinations of eta quotients may even be integral. For example, define,

<math>\begin{align}

j(\tau)&=\left(\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{8}+2^8 \left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{16}\right)^3 \\[6pt]

j_{2A}(\tau)&=\left(\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{12}+2^6 \left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{12}\right)^2 \\[6pt]

j_{3A}(\tau) &=\left(\left(\frac{\eta(\tau)}{\eta(3\tau)}\right)^{6}+3^3 \left(\frac{\eta(3\tau)}{\eta(\tau)}\right)^{6}\right)^2 \\[6pt]

j_{4A}(\tau) &=\left(\left(\frac{\eta(\tau)}{\eta(4\tau)}\right)^{4} + 4^2 \left(\frac{\eta(4\tau)}{\eta(\tau)}\right)^{4}\right)^2 = \left(\frac{\eta^2(2\tau)}{\eta(\tau)\,\eta(4\tau)} \right)^{24} \end{align}</math>

with the 24th power of the Weber modular function Template:Math. Then,

<math>\begin{align}

j\left(\frac{1+\sqrt{-163}}{2}\right) &= -640320^3, & e^{\pi\sqrt{163}} &\approx 640320^3+743.99999999999925\dots \\[6pt]

j_{2A}\left(\frac{\sqrt{-58}}{2}\right) &= 396^4, & e^{\pi\sqrt{58}}&\approx 396^4-104.00000017\dots \\[6pt]

j_{3A}\left(\frac{1+\sqrt{-\frac{89}{3}}}{2}\right) &= -300^3, & e^{\pi\sqrt\frac{89}{3}}&\approx 300^3+41.999971\dots \\[6pt]

j_{4A}\left(\frac{\sqrt{-7}}{2}\right)&=2^{12}, & e^{\pi\sqrt{7}}&\approx 2^{12}-24.06\dots \end{align}</math>

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

Eta quotients may also be a useful tool for describing bases of modular forms, which are notoriously difficult to compute and express directly. In 1993 Basil Gordon and Kim Hughes proved that if an eta quotient Template:Mvar of the form given above, namely <math> \prod_{0<d\mid N}\eta(d\tau)^{r_d} </math> satisfies

<math> \sum_{0<d\mid N}d r_d \equiv 0 \pmod{24} \quad \text{and} \quad \sum_{0<d\mid N}\frac{N}{d}r_d \equiv 0 \pmod{24},</math>

then Template:Mvar is a [[modular form|weight Template:Mvar modular form]] for the congruence subgroup Template:Math (up to holomorphicity) where<ref>Template:Cite book</ref>

<math>k=\frac12\sum_{0<d\mid N} r_d.</math>

This result was extended in 2019 such that the converse holds for cases when Template:Mvar is coprime to 6, and it remains open that the original theorem is sharp for all integers Template:Mvar.<ref name="AAHOS">Template:Cite journal</ref> This also extends to state that any modular eta quotient for any [[Congruence subgroup| level Template:Mvar congruence subgroup]] must also be a modular form for the group Template:Math. While these theorems characterize modular eta quotients, the condition of holomorphicity must be checked separately using a theorem that emerged from the work of Gérard Ligozat<ref>Template:Cite book</ref> and Yves Martin:<ref>Template:Cite journal</ref>

If Template:Mvar is an eta quotient satisfying the above conditions for the integer Template:Mvar and Template:Mvar and Template:Mvar are coprime integers, then the order of vanishing at the cusp Template:Math relative to Template:Math is

<math>\frac{N}{24}\sum_{0<\delta|N} \frac{\gcd\left(d,\delta\right)^2r_\delta}{\gcd\left(d,\frac{N}{\delta}\right)d\delta} .</math>

These theorems provide an effective means of creating holomorphic modular eta quotients, however this may not be sufficient to construct a basis for a vector space of modular forms and cusp forms. A useful theorem for limiting the number of modular eta quotients to consider states that a holomorphic weight Template:Mvar modular eta quotient on Template:Math must satisfy

<math>\sum_{0<d\mid N} |r_d|\leq \prod_{p\mid N}\left(\frac{p+1}{p-1}\right)^{\min\bigl(2,\text{ord}_p(N)\bigr)},</math>

where Template:Math denotes the largest integer Template:Mvar such that Template:Mvar divides Template:Mvar.<ref name="RW">Template:Cite journal</ref> These results lead to several characterizations of spaces of modular forms that can be spanned by modular eta quotients.<ref name="RW"/> Using the graded ring structure on the ring of modular forms, we can compute bases of vector spaces of modular forms composed of Template:Math-linear combinations of eta-quotients. For example, if we assume Template:Math is a semiprime then the following process can be used to compute an eta-quotient basis of [[modular form|Template:Math]].<ref name="AAHOS" />

Template:Ordered list A collection of over 6300 product identities for the Dedekind eta function in a canonical, standardized form is available at the Wayback machine<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> of Michael Somos' website.

See also

• Chowla–Selberg formula
• Ramanujan–Sato series
• q-series
• Weierstrass elliptic function
• Partition function
• Kronecker limit formula
• Affine Lie algebra

References

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Further reading

• Template:Cite book
• Template:Cite book