Inmathematics, theDedekind eta function, named afterRichard Dedekind, is amodular form of weight 1/2 and is a function defined on theupper half-plane ofcomplex numbers, where the imaginary part is positive. It also occurs inbosonic string theory.

For any complex numberτ withIm(τ) > 0, letq =e^2πiτ; then the eta function is defined by,

${\displaystyle \eta (\tau )=e^{\frac{\pi i\tau }{12}}\prod _{n=1}^{\mathrm{\infty }}\left(1-e^{2n\pi i\tau }\right)=q^{\frac{1}{24}}\prod _{n=1}^{\mathrm{\infty }}\left(1-q^n\right).}$

Raising the eta equation to the 24th power and multiplying by(2π)¹² gives

${\displaystyle \mathrm{\Delta }(\tau )=(2\pi {)}^{12}{\eta }^{24}(\tau )}$

whereΔ is themodular discriminant. The presence of24 can be understood by connection with other occurrences, such as in the 24-dimensionalLeech lattice.

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

The eta function satisfies thefunctional equations^[1]

In the second equation thebranch of the square root is chosen such that√−iτ = 1 whenτ =i.

More generally, supposea,b,c,d are integers withad −bc = 1, so that

${\displaystyle \tau \mapsto \frac{a\tau +b}{c\tau +d}}$

is a transformation belonging to themodular group. We may assume that eitherc > 0, orc = 0 andd = 1. Then

${\displaystyle \eta \left(\frac{a\tau +b}{c\tau +d}\right)=ϵ(a,b,c,d){\left(c\tau +d\right)}^{\frac{1}{2}}\eta (\tau ),}$

where

Heres(h,k) is theDedekind sum

${\displaystyle s(h,k)=\sum _{n=1}^{k-1}\frac{n}{k}\left(\frac{hn}{k}-\lfloor \frac{hn}{k}\rfloor -\frac{1}{2}\right).}$

Because of these functional equations the eta function is amodular form of weight⁠1/2⁠ and level 1 for a certain character of order 24 of themetaplectic double cover of the modular group, and can be used to define other modular forms. In particular themodular discriminant of theWeierstrass elliptic function with

${\displaystyle {\omega }_2=\tau {\omega }_1}$

can be defined as

${\displaystyle \mathrm{\Delta }(\tau )=(2\pi {\omega }_1{)}^{12}\eta (\tau {)}^{24}}$

and is a modular form of weight 12. Some authors omit the factor of(2π)¹², so that the series expansion has integral coefficients.

TheJacobi triple product implies that the eta is (up to a factor) a Jacobitheta function for special values of the arguments:^[2]

${\displaystyle \eta (\tau )=\sum _{n=1}^{\mathrm{\infty }}\chi (n)\exp \left(\frac{\pi i{n}^2\tau }{12}\right),}$

whereχ(n) is "the"Dirichlet character modulo 12 withχ(±1) = 1 andχ(±5) = −1. Explicitly,^{[citation needed]}

${\displaystyle \eta (\tau )=e^{\frac{\pi i\tau }{12}}\vartheta \left(\frac{\tau +1}{2};3\tau \right).}$

TheEuler function

has a power series by theEuler identity:

${\displaystyle \varphi (q)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^n{q}^{\frac{3n^2-n}{2}}.}$

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

${\displaystyle \eta (\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\pi in}{e}^{3\pi i{\left(n+\frac{1}{6}\right)}^2\tau }.}$

This can be proved by using${\displaystyle x=2\pi i\tau }$ in Euler Pentagonal number theorem with the definition of eta function.

Because the eta function is easy to compute numerically from eitherpower 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 ofq^⁠1/24⁠ between this and eta makes almost no visual difference whatsoever. Thus, this picture can be taken as a picture of eta as a function ofq.

The theory of thealgebraic characters of theaffine Lie algebras gives rise to a large class of previously unknown identities for the eta function. These identities follow from theWeyl–Kac character formula, and more specifically from the so-called "denominator identities". The characters themselves allow the construction of generalizations of theJacobi theta function which transform under themodular group; this is what leads to the identities. An example of one such new identity^[3] is

${\displaystyle \eta (8\tau )\eta (16\tau )=\sum _{\genfrac{}{}{0ex}{}{m,n\in \mathbb{Z}}{m\le |3n|}}(-1{)}^m{q}^{(2m+1{)}^2-32n^2}}$

whereq =e^2πiτ is theq-analog or "deformation" of thehighest weight of a module.

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

Eta quotients are defined by quotients of the form

whered is a non-negative integer andr_d is any integer. Linear combinations of eta quotients at imaginary quadratic arguments may bealgebraic, while combinations of eta quotients may even beintegral. For example, define,

with the 24th power of theWeber modular function𝔣(τ). Then,

and so on, values which appear inRamanujan–Sato series.

Eta quotients may also be a useful tool for describing bases ofmodular forms, which are notoriously difficult to compute and express directly. In 1993 Basil Gordon and Kim Hughes proved that if an eta quotientη_g of the form given above, namely satisfies

thenη_g is aweightk modular form for thecongruence subgroupΓ₀(N) (up toholomorphicity) where^[4]

This result was extended in 2019 such that the converse holds for cases whenN iscoprime to 6, and it remains open that the original theorem is sharp for all integersN.^[5] This also extends to state that anymodular eta quotient for any leveln congruence subgroup must also be a modular form for the groupΓ(N). While these theorems characterizemodular eta quotients, the condition of holomorphicity must be checked separately using a theorem that emerged from the work of Gérard Ligozat^[6] and Yves Martin:^[7]

Ifη_g is an eta quotient satisfying the above conditions for the integerN andc andd are coprime integers, then the order of vanishing at the cusp⁠c/d⁠ relative toΓ₀(N) is

These theorems provide an effective means of creating holomorphic modular eta quotients, however this may not be sufficient to construct a basis for avector space of modular forms andcusp forms. A useful theorem for limiting the number of modular eta quotients to consider states that a holomorphic weightk modular eta quotient onΓ₀(N) must satisfy

whereord_p(N) denotes the largest integerm such thatp^m dividesN.^[8]These results lead to several characterizations of spaces of modular forms that can be spanned by modular eta quotients.^[8] Using thegraded ring structure on the ring of modular forms, we can compute bases of vector spaces of modular forms composed of${\displaystyle \mathbb{C}}$-linear combinations of eta-quotients. For example, if we assumeN =pq is asemiprime then the following process can be used to compute an eta-quotient basis ofM_k(Γ₀(N)).^[5]

A collection of over 6300 product identities for the Dedekind Eta Function in a canonical, standardized form is available at the Wayback machine^[9] of Michael Somos' website.

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

• Apostol, Tom M. (1990).Modular functions and Dirichlet Series in Number Theory.Graduate Texts in Mathematics. Vol. 41 (2nd ed.). Springer-Verlag. ch. 3.ISBN 3-540-97127-0.
• Koblitz, Neal (1993).Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics. Vol. 97 (2nd ed.). Springer-Verlag.ISBN 3-540-97966-2.

• Fractals
• Modular forms
• Elliptic functions

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