In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Dedekind eta function, named after
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
, is a
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
of weight 1/2 and is a function defined on the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, where the imaginary part is positive. It also occurs in
bosonic string theory
Bosonic string theory is the original version of string theory, developed in the late 1960s and named after Satyendra Nath Bose. It is so called because it contains only bosons in the spectrum.
In the 1980s, supersymmetry was discovered in the c ...
.
Definition
For any complex number with , let ; then the eta function is defined by,
:
Raising the eta equation to the 24th power and multiplying by gives
:
where is the
modular discriminant
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the ...
. The presence of
24 can be understood by connection with other occurrences, such as in the 24-dimensional
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by ...
.
The eta function is
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
on the upper half-plane but cannot be continued analytically beyond it.
![Q-Eulero](https://upload.wikimedia.org/wikipedia/commons/f/f4/Q-Eulero.jpeg)
The eta function satisfies the
functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
s
:
In the second equation the
branch of the square root is chosen such that when .
More generally, suppose are integers with , so that
:
is a transformation belonging to the
modular group. We may assume that either , or and . Then
:
where
:
Here is the
Dedekind sum In mathematics, Dedekind sums are certain sums of products of a sawtooth function, and are given by a function ''D'' of three integer variables. Dedekind introduced them to express the functional equation of the Dedekind eta function. They have su ...
:
Because of these functional equations the eta function is a
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
of weight 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
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the ...
of
Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university without a degree, ...
can be defined as
:
and is a modular form of weight 12. Some authors omit the factor of , 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
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
for special values of the arguments:
:
where is "the"
Dirichlet character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b:
:1) \chi ...
modulo 12 with and . Explicitly,
:
The
Euler function
In mathematics, the Euler function is given by
:\phi(q)=\prod_^\infty (1-q^k),\quad , q, A000203
On account of the identity \sum_ d = \sum_ \frac, this may also be written as
:\ln(\phi(q)) = -\sum_^\infty \frac \sum_ d.
Also if a,b\in\mathbb^ ...
:
has a power series by the
Euler identity:
:
Because the eta function is easy to compute numerically from either
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
, 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 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 .
Combinatorial identities
The theory of the
algebraic character
An algebraic character is a formal expression attached to a module in representation theory of semisimple Lie algebras that generalizes the character of a finite-dimensional representation and is analogous to the Harish-Chandra character of the re ...
s of the
affine Lie algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...
s gives rise to a large class of previously unknown identities for the eta function. These identities follow from the
Weyl–Kac character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the cha ...
, and more specifically from the so-called "denominator identities". The characters themselves allow the construction of generalizations of the
Jacobi theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
which transform under the
modular group; this is what leads to the identities. An example of one such new identity is
:
where is the
-analog or "deformation" of the
highest weight In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multipli ...
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
:
Eta quotients
Eta quotients are defined by quotients of the form
:
where is a non-negative integer and is any integer. Linear combinations of eta quotients at imaginary quadratic arguments may be
algebraic, while combinations of eta quotients may even be
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
. For example, define,
:
with the 24th power of the
Weber modular function In mathematics, the Weber modular functions are a family of three functions ''f'', ''f''1, and ''f''2,''f'', ''f''1 and ''f''2 are not Modular form#Modular functions, modular functions (per the Wikipedia definition), but every modular function is a ...
. Then,
:
and so on, values which appear in
Ramanujan–Sato series In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as,
:\frac = \frac \sum_^\infty \frac \frac
to the form
:\frac = \sum_^\infty s(k) \frac
by using other well-defined sequences of integers s(k) obeying a cer ...
.
Eta quotients may also be a useful tool for describing bases of
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
s, which are notoriously difficult to compute and express directly. In 1993 Basil Gordon and Kim Hughes proved that if an eta quotient of the form given above, namely
satisfies
:
then is a
weight modular form for the
congruence subgroup
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the ...
(up to
holomorphicity) where
:
This result was extended in 2019 such that the converse holds for cases when is
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
to 6, and it remains open that the original theorem is sharp for all integers .
This also extends to state that any
modular eta quotient for any
level congruence subgroup must also be a modular form for the group . While these theorems characterize
modular
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
eta quotients, the condition of
holomorphicity must be checked separately using a theorem that emerged from the work of Gérard Ligozat and Yves Martin:
If is an eta quotient satisfying the above conditions for the integer and and are coprime integers, then the order of vanishing at the
cusp relative to is
:
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
of modular forms and
cusp forms. A useful theorem for limiting the number of modular eta quotients to consider states that a holomorphic weight modular eta quotient on must satisfy
:
where denotes the largest integer such that divides .
These results lead to several characterizations of spaces of modular forms that can be spanned by modular eta quotients.
Using the
graded ring
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
structure on the ring of modular forms, we can compute bases of vector spaces of modular forms composed of -linear combinations of eta-quotients. For example, if we assume is a
semiprime
In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers.
Because there are infinitely many prime nu ...
then the following process can be used to compute an eta-quotient basis of
.
A collection of over 6300 product identities for the Dedekind Eta Function in a canonical, standardized form is available at the Wayback machine
of Michael Somos' website.
See also
*
Chowla–Selberg formula
In mathematics, the Chowla–Selberg formula is the evaluation of a certain product of values of the gamma function at rational values in terms of values of the Dedekind eta function at imaginary quadratic irrational numbers. The result was essent ...
*
Ramanujan–Sato series In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as,
:\frac = \frac \sum_^\infty \frac \frac
to the form
:\frac = \sum_^\infty s(k) \frac
by using other well-defined sequences of integers s(k) obeying a cer ...
*
q-series
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product
(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^),
with (a;q)_0 = 1.
It is a ''q''-analog of the Pochhammer sym ...
*
Weierstrass's elliptic functions
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by t ...
*
Partition function
*
Kronecker limit formula In mathematics, the classical Kronecker limit formula describes the constant term at ''s'' = 1 of a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There are many generalizations of it to more comp ...
*
Affine Lie algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...
References
Further reading
*
* {{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
Fractals
Modular forms
Elliptic functions