Editing Dedekind eta function

{{Short description|Mathematical function}}
{{distinguish|Weierstrass eta function|Dirichlet eta function}}
[[Image:Dedekind Eta.jpg|right|thumb|500px|Dedekind {{mvar|η}}-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 number]]s, where the imaginary part is positive. It also occurs in [[bosonic string theory]].

==Definition==
For any complex number {{mvar|τ}} with {{math|Im(''τ'') > 0}}, let {{math|''q'' {{=}} ''e''<sup>2''πiτ''</sup>}}; 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 {{math|(2''π'')<sup>12</sup>}} gives

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

where {{math|Δ}} is the [[modular discriminant]].  The presence of [[24 (number)|24]] can be understood by connection with other occurrences, such as in the 24-dimensional [[Leech lattice]].

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

[[File:Q-Eulero.jpeg|thumb|right|Modulus of Euler phi on the unit disc, colored so that black = 0, red = 4]]
[[Image:Discriminant real part.jpeg|thumb|right|The real part of the modular discriminant as a function of {{mvar|q}}.]]

The eta function satisfies the [[functional equation]]s<ref>{{cite journal|last=Siegel|first=C. L.|title=A Simple Proof of ''η''(−1/''τ'') {{=}} ''η''(''τ''){{sqrt|''τ''/''i''}}|journal=[[Mathematika]]|year=1954|volume=1|page=4|doi=10.1112/S0025579300000462}}</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 [[Complex square root|branch of the square root]] is chosen such that {{math|{{sqrt|−''iτ''}} {{=}} 1}} when {{math|''τ'' {{=}} ''i''}}.

More generally, suppose {{math|''a'', ''b'', ''c'', ''d''}} are integers with {{math|''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 {{math|''c'' > 0}}, or {{math|''c'' {{=}} 0}} and {{math|''d'' {{=}} 1}}.  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 {{math|''s''(''h'',''k'')}} 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 {{sfrac|1|2}} and level 1 for a certain character of order 24 of the [[metaplectic group|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 {{math|(2''π'')<sup>12</sup>}}, 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>{{citation|first=Daniel|last= Bump|title=Automorphic Forms and Representations|year=1998|publisher=Cambridge University Press|isbn=0-521-55098-X}}</ref>

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

where {{math|''χ''(''n'')}} is "the" [[Dirichlet character]] modulo 12 with {{math|''χ''(±1) {{=}} 1}} and {{math|''χ''(±5) {{=}} −1}}. Explicitly,{{Citation needed|date=September 2016}}

:<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 [[Pentagonal number theorem|Euler identity]]:

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

Note that by using [[Pentagonal number theorem| 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 [[Pentagonal number theorem| 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 {{math|''q''<sup>{{sfrac|1|24}}</sup>}} 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 {{mvar|q}}.

== Combinatorial identities ==
The theory of the [[algebraic character]]s of the [[affine Lie algebra]]s 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>{{citation|first=Jurgen|last= Fuchs|title=Affine Lie Algebras and Quantum Groups|year=1992|publisher=Cambridge University Press|isbn=0-521-48412-X}}</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 {{math|''q'' {{=}} ''e''<sup>2''πiτ''</sup>}} is the [[q-analog|{{mvar|q}}-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 {{mvar|d}} is a non-negative integer and {{mvar|r<sub>d</sub>}} is any integer. Linear combinations of eta quotients at imaginary quadratic arguments may be [[algebraic number|algebraic]], while combinations of eta quotients may even be [[integer|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]] {{math|𝔣(''τ'')}}. 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 form]]s, which are notoriously difficult to compute and express directly. In 1993 Basil Gordon and Kim Hughes proved that if an eta quotient {{mvar|η<sub>g</sub>}} 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 {{mvar|η<sub>g</sub>}} is a [[modular form|weight {{mvar|k}} modular form]] for the [[congruence subgroup]] {{math|Γ<sub>0</sub>(''N'')}} (up to [[Holomorphic function|holomorphicity]]) where<ref>{{cite book|first1=Basil |last1=Gordon |first2=Kim |last2=Hughes |contribution=Multiplicative properties of ''η''-products. II. |title=A Tribute to Emil Grosswald: Number Theory and Related Analysis |volume=143 |series=Contemporary Mathematics |page=415–430 |publisher=American Mathematical Society |location=Providence, RI |date=1993}}</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 {{mvar|N}} is [[Coprime integers|coprime]] to 6, and it remains open that the original theorem is sharp for all integers {{mvar|N}}.<ref name="AAHOS">{{cite journal|first1=Michael |last1=Allen|first2=Nicholas |last2=Anderson|first3=Asimina |last3=Hamakiotes|first4=Ben |last4=Oltsik|first5=Holly |last5=Swisher|title=Eta-quotients of prime or semiprime level and elliptic curves|journal=Involve|year=2020|volume=13|issue=5|pages=879–900 |doi=10.2140/involve.2020.13.879|arxiv=1901.10511|s2cid=119620241 }}</ref>  This also extends to state that any [[modular form|modular eta quotient]] for any [[Congruence subgroup| level {{mvar|n}} congruence subgroup]] must also be a modular form for the group {{math|Γ(''N'')}}. While these theorems characterize [[Modular form|modular]] eta quotients, the condition of [[Holomorphic function| holomorphicity]] must be checked separately using a theorem that emerged from the work of Gérard Ligozat<ref>{{cite book|first=G. |last=Ligozat |title=Courbes modulaires de genre 1 |publisher=U.E.R. Mathématique, Université Paris XI, Orsay |date=1974 |series=Publications Mathématiques d'Orsay |volume=75 |page=7411}}</ref> and Yves Martin:<ref>{{cite journal|first=Yves |last=Martin|title=Multiplicative ''η''-quotients|journal=[[Transactions of the American Mathematical Society]]|year=1996|volume=348|issue=12|page=4825–4856|doi=10.1090/S0002-9947-96-01743-6 |doi-access=free}}</ref>

If {{mvar|η<sub>g</sub>}} is an eta quotient satisfying the above conditions for the integer {{mvar|N}} and {{mvar|c}} and {{mvar|d}} are coprime integers, then the order of vanishing at the [[Cusp (singularity)| cusp]] {{math|{{sfrac|''c''|''d''}}}} relative to {{math|Γ<sub>0</sub>(''N'')}} 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 form|cusp forms]]. A useful theorem for limiting the number of modular eta quotients to consider states that a holomorphic weight {{mvar|k}} modular eta quotient on {{math|Γ<sub>0</sub>(''N'')}} 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 {{math|ord<sub>''p''</sub>(''N'')}} denotes the largest integer {{mvar|m}} such that {{mvar|p<sup>m</sup>}} divides {{mvar|N}}.<ref name="RW">{{cite journal|first1=Jeremy |last1=Rouse|first2=John J. |last2=Webb|title=On spaces of modular forms spanned by eta-quotients|journal=[[Advances in Mathematics]]|year=2015|volume=272|page=200–224|doi=10.1016/j.aim.2014.12.002|doi-access=free|arxiv=1311.1460}}</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 {{math|<math>\mathbb{C}</math>}}-linear combinations of eta-quotients. For example, if we assume {{math|''N'' {{=}} ''pq''}} is a [[semiprime]] then the following process can be used to compute an eta-quotient basis of [[modular form|{{math|''M<sub>k</sub>''(Γ<sub>0</sub>(''N''))}}]].<ref name="AAHOS" />

{{ordered list
|Fix a semiprime {{math|''N'' {{=}} ''pq''}} which is coprime to 6 (that is, {{math|''p'', ''q'' > 3}}). We know that any modular eta quotient may be found using the above theorems, therefore it is reasonable to algorithmically to compute them.

|Compute the dimension {{mvar|D}} of {{math|''M<sub>k</sub>''(Γ<sub>0</sub>(''N''))}}. This tells us how many linearly-independent modular eta quotients we will need to compute to form a basis.

|Reduce the number of eta quotients to consider. For semiprimes we can reduce the number of partitions using the bound on
:<math>\sum_{0<d\mid N} |r_d|</math> 
and by noticing that the sum of the orders of vanishing at the cusps of {{math|Γ<sub>0</sub>(''N'')}} must equal
:<math>S:=\frac{(p+1)(q+1)}{6}</math>.<ref name="AAHOS"/>

|Find all partitions of {{mvar|S}} into 4-tuples (there are 4 cusps of {{math|Γ<sub>0</sub>(''N'')}}), and among these consider only the partitions which satisfy Gordon and Hughes' conditions (we can convert orders of vanishing into exponents). Each of these partitions corresponds to a unique eta quotient.

|Determine the minimum number of terms in the [[modular form|{{mvar|q}}-expansion]] of each eta quotient required to identify elements uniquely (this uses a result known as [[Sturm's bound]]). Then use linear algebra to determine a maximal independent set among these eta quotients.

|Assuming that we have not already found {{mvar|D}} linearly independent eta quotients, find an appropriate vector space {{math|''M''<sub>''k''{{prime}}</sub>(Γ<sub>0</sub>(''N''))}} such that {{math|''k''{{prime}}}} and {{math|''M''<sub>''k''{{prime}}</sub>(Γ<sub>0</sub>(''N''))}} is spanned by ([[Weakly holomorphic modular form|weakly holomorphic]]) eta quotients,<ref name="RW"/> and {{math|''M''<sub>''k''{{prime}}−''k''</sub>(Γ<sub>0</sub>(''N''))}} contains an eta quotient {{mvar|η<sub>g</sub>}}.

|Take a modular form {{mvar|f}} with weight {{mvar|k}} that is not in the span of our computed eta quotients, and compute {{math|''f'' ''η<sub>g</sub>''}} as a linear combination of eta-quotients in {{math|''M''<sub>''k''{{prime}}</sub>(Γ<sub>0</sub>(''N''))}} and then divide out by {{mvar|η<sub>g</sub>}}. The result will be an expression of {{mvar|f}} as a linear combination of eta quotients as desired. Repeat this until a basis is formed.
}}
A collection of over 6300 product identities for the Dedekind eta function in a canonical, standardized form is available at the Wayback machine<ref>{{cite web | url=http://eta.math.georgetown.edu/index.html | archive-url=https://web.archive.org/web/20190709153048/http://eta.math.georgetown.edu/index.html | archive-date=2019-07-09 | title=Dedekind Eta Function Product Identities by Michael Somos }}</ref> of Michael Somos' website.

==See also==

* [[Chowla–Selberg formula]]
* [[Ramanujan–Sato series#Level 2|Ramanujan–Sato series]]
* [[q-series]]
* [[Weierstrass elliptic function]]
* [[Partition function (number theory)|Partition function]]
* [[Kronecker limit formula]]
* [[Affine Lie algebra]]

==References==
<references/>

==Further reading==
* {{cite book|first=Tom M. |last=Apostol |title=Modular functions and Dirichlet Series in Number Theory |edition=2nd |series=[[Graduate Texts in Mathematics]] |volume=41 |date=1990 |publisher=Springer-Verlag |isbn=3-540-97127-0 |at=ch. 3}}
* {{cite book|first=Neal |last=Koblitz |authorlink=Neal Koblitz |title=Introduction to Elliptic Curves and Modular Forms |edition=2nd |series=Graduate Texts in Mathematics |volume=97 |date=1993 |publisher=Springer-Verlag |isbn=3-540-97966-2}}

[[Category:Fractals]]
[[Category:Modular forms]]
[[Category:Elliptic functions]]