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Mathematical function
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
Dedekind_eta_function
Mathematical identities related to integer partitions
{1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}} The Dedekind eta function identities for the functions G and H result by combining only the following two
Rogers–Ramanujan_identities
Topics referred to by the same term
theory, Dedekind function can refer to any of three functions, all introduced by Richard Dedekind Dedekind eta function Dedekind psi function Dedekind zeta
Dedekind_function
Evaluates a certain product of values of the Gamma function at rational values
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
Chowla–Selberg_formula
Seventh letter in the Greek alphabet
lambda calculus. Mathematics, the Dirichlet eta function, Dedekind eta function, and Weierstrass eta function. In category theory, the unit of an adjunction
Eta
Function studied by Ramanujan
(q)^{24}=\eta (z)^{24}=\Delta (z),} where ϕ {\displaystyle \phi } is the Euler function, η {\displaystyle \eta } is the Dedekind eta function, Δ ( z )
Ramanujan_tau_function
Mathematical functions related to Weierstrass's elliptic function
Weierstrass eta function should not be confused with either the Dedekind eta function or the Dirichlet eta function. The Weierstrass p-function is related
Weierstrass_functions
Mathematical theorem about the real analytic Eisenstein series
a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There are many generalizations of it to more complicated
Kronecker_limit_formula
Special functions of several complex variables
n − 1 , {\displaystyle \tau =n{\sqrt {-1}},} and Dedekind eta function η ( τ ) . {\displaystyle \eta (\tau ).} Then for n = 1 , 2 , 3 , … {\displaystyle
Theta_function
introduced them in the 1880's to express the functional equation of the Dedekind eta function, in a commentary to Bernhard Riemann's collected papers.. They have
Dedekind_sum
Natural number
modular forms through the Dedekind eta function η ( τ ) = q 1 / 24 ∏ n > 0 ( 1 − q n ) , q = e 2 π i τ . {\displaystyle \eta (\tau )=q^{1/24}\prod _{n>0}(1-q^{n})
24_(number)
German mathematician (1831–1916)
Richard Dedekind Dedekind cut Dedekind domain Dedekind eta function Dedekind-infinite set Dedekind number Dedekind psi function Dedekind sum Dedekind zeta
Richard_Dedekind
Topics referred to by the same term
eta function may refer to: The Dirichlet eta function η(s), a Dirichlet series The Dedekind eta function η(τ), a modular form The Weierstrass eta function
Eta_function
Modular Functions". The function η ( τ ) {\displaystyle \eta (\tau )} is the Dedekind eta function and ( e 2 π i τ ) α {\displaystyle (e^{2\pi i\tau })^{\alpha
Weber_modular_function
Modular function in mathematics
)^{3}-27g_{3}(\tau )^{2}=(2\pi )^{12}\,\eta (\tau )^{24}} , Dedekind eta function η ( τ ) {\displaystyle \eta (\tau )} , and modular invariants, g 2 (
J-invariant
Mathematical function
{\displaystyle (3n^{2}-n)/2} is a pentagonal number. The Euler function is related to the Dedekind eta function as ϕ ( e 2 π i τ ) = e − π i τ / 12 η ( τ ) . {\displaystyle
Euler_function
functions Theta functions Neville theta functions Modular lambda function Closely related are the modular forms, which include J-invariant Dedekind eta
List of mathematical functions
List_of_mathematical_functions
Number of partitions of an integer
specifically the Dedekind eta function. The same sequence of pentagonal numbers appears in a recurrence relation for the partition function: p ( n ) = ∑ k
Partition function (number theory)
Partition_function_(number_theory)
Indian mathematician (1887–1920)
(\theta )|<\pi } , where Γ(z) is the gamma function, and related to a special value of the Dedekind eta function. Expanding into series of powers and equating
Srinivasa_Ramanujan
Mathematical function
Euler function, which is closely related to the Dedekind eta function. The Jacobi theta function may be written in terms of the Ramanujan theta function as:
Ramanujan_theta_function
Symmetric holomorphic function
(\tau )=k^{2}(\tau )} . In terms of the Dedekind eta function η ( τ ) {\displaystyle \eta (\tau )} and theta functions, λ ( τ ) = ( 2 η ( τ 2 ) η 2 ( 2 τ )
Modular_lambda_function
relation Cyclotomic polynomials H. G. Dawson: Dawson function Richard Dedekind: Dedekind eta function Charles F. Dunkl: Dunkl operator, Jacobi–Dunkl operator
List of eponyms of special functions
List_of_eponyms_of_special_functions
Continued fraction closely related to the Rogers–Ramanujan identities
throughout this section since the q-expansion of the j-function (as well as the well-known Dedekind eta function) uses q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau
Rogers–Ramanujan continued fraction
Rogers–Ramanujan_continued_fraction
Sporadic simple group
{3}+12256q^{4}+39350q^{5}+\dots \end{aligned}}} and η(τ) is the Dedekind eta function. Norton & Wilson (1986) found the 14 conjugacy classes of maximal
Harada–Norton_group
axiom Dedekind completeness Dedekind cut Dedekind discriminant theorem Dedekind domain Dedekind eta function Dedekind function Dedekind group Dedekind number
List of things named after Richard Dedekind
List_of_things_named_after_Richard_Dedekind
Analytic function on the upper half-plane with a certain behavior under the modular group
discriminant The Dedekind eta function is defined as η ( z ) = q 1 / 24 ∏ n = 1 ∞ ( 1 − q n ) , q = e 2 π i z . {\displaystyle \eta (z)=q^{1/24}\prod
Modular_form
Sporadic simple group
^{3}+4160q^{4}+13015q^{5}+\dots \end{aligned}}} and η(τ) is the Dedekind eta function. Aschbacher, Michael (1997), 3-transposition groups, Cambridge Tracts
Fischer_group_Fi22
Class of mathematical functions
24 {\displaystyle \Delta =(2\pi )^{12}\eta ^{24}} where η {\displaystyle \eta } is the Dedekind eta function. For the Fourier coefficients of Δ {\displaystyle
Weierstrass_elliptic_function
Natural number
Ramanujan τ {\displaystyle \tau } -function and which is (up to a constant multiplier) the 24th power of the Dedekind eta function: Δ ( τ ) = ( 2 π ) 12 η 24
12_(number)
z ) {\displaystyle \eta (z)} denote the Dedekind eta function. Then for q = e 2 π i z {\displaystyle q=e^{2\pi iz}} , the function S ~ ( z ) := q − 1 /
Spt_function
Plane algebraic curve
x with coefficients in Z[y], it has degree ψ(n), where ψ is the Dedekind psi function. Since Φn(x, y) = Φn(y, x), X0(n) is symmetrical around the line
Classical_modular_curve
Type of generalization of periodic functions in Euclidean space
harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G {\displaystyle G} to the complex numbers (or
Automorphic_form
Symbols for constants, special functions
an effect size measure for analyses of variance the eta meson viscosity the Dedekind eta function energy conversion efficiency efficiency (physics) the
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Series related to Ramanujan's pi formulas
\end{aligned}}} with the j-function j(τ), Eisenstein series E4, and Dedekind eta function η(τ). The first expansion is the McKay–Thompson
Ramanujan–Sato_series
Sporadic simple group
{2}+11202q^{3}+49152q^{4}+\dots \end{aligned}}} and η(τ) is the Dedekind eta function. Conway et al. (1985) "ATLAS: Conway group Co3". "ATLAS: Conway
Conway_group_Co3
Sporadic simple group
40002q^{3}+10698752q^{4}+\cdots \end{aligned}}} and η(τ) is the Dedekind eta function. Wilson (1999) found the 30 conjugacy classes of maximal subgroups
Baby_monster_group
Modular form
of the Dedekind eta function. The Fourier coefficients here are written τ ( n ) {\displaystyle \tau (n)} and called 'Ramanujan's tau function', with the
Cusp_form
Mathematical conjecture about zeros of L-functions
L-functions than Dedekind zeta functions lie on critical lines. One example can be Ramanujan L-function related to modular form called Dedekind eta function
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
26-dimensional string theory
_{1}<{\frac {1}{2}}\right\}} . η ( τ ) {\displaystyle \eta (\tau )} is the Dedekind eta function. The integrand is of course invariant under the modular
Bosonic_string_theory
Concept in algebraic number theory
the cubics can be exactly given by quotients of the Dedekind eta function η(τ), a modular function involving a 24th root, and which explains the 24 in
Heegner_number
Dedekind eta function and the modular discriminant, which connection is deepened by Monstrous moonshine, a development that related modular functions
Exceptional_object
Type of Kac–Moody algebras
identities, include many previously unknown identities for the Dedekind eta function. These generalizations can be viewed as a practical example of the
Affine_Lie_algebra
Prime number with a certain relationship to an elliptic curve
{\displaystyle \eta (\tau )^{2}\eta (11\tau )^{2}} vanishes modulo p {\displaystyle p} , where η {\displaystyle \eta } is the Dedekind eta function. More generally
Supersingular prime (algebraic number theory)
Supersingular_prime_(algebraic_number_theory)
Varying methods used to calculate pi
two, but this time is a quotient of a modular form, namely the Dedekind eta function, and where the argument involves τ = − 3502 {\displaystyle \tau
Approximations_of_pi
Algebra describing 2D conformal symmetry
{c-1}{24}}}}{\eta (q)}}=q^{h-{\frac {c}{24}}}\left(1+q+2q^{2}+3q^{3}+5q^{4}+\cdots \right),} where η {\displaystyle \eta } is the Dedekind eta function. For any
Virasoro_algebra
Algebraic curve in mathematics
)^{2}=(2\pi )^{12}\,\eta ^{24}(\tau )} is generally a transcendental number. In particular, the value of the Dedekind eta function η(2i) is η ( 2 i ) =
Elliptic_curve
Sporadic simple group
q^{3}+1956q^{4}+5135q^{5}+\dots \end{aligned}}} and η(τ) is the Dedekind eta function. It can be defined in terms of the generators a and b and relations
Held_group
Sporadic simple group
2}+65367q^{3}+371520q^{4}+\dots \end{aligned}}} and η(τ) is the Dedekind eta function. Kleidman, Parker & Wilson (1989) found the 14 conjugacy classes
Fischer_group_Fi23
{\displaystyle \eta (z)} the Dedekind eta function, the modular form f α , β ( z ) = η ( z ) 2 α η ( z ) 2 β ¯ ¯ {\displaystyle f_{\alpha ,\beta }(z)=\eta (z)^{2\alpha
Automorphic_factor
British-Lebanese mathematician (1929–2019)
Segal on twisted K-theory. One paper is a detailed study of the Dedekind eta function from the point of view of topology and the index theorem. Several
Michael_Atiyah
Four finite groups derived from the Leech lattice
)}{\eta (\tau )}}\right)^{4}\right)^{2}\\&={\frac {1}{q}}+24+276q+2048q^{2}+11202q^{3}+49152q^{4}+\dots \end{aligned}}} and η(τ) is the Dedekind eta function
Conway_group
Theorem in number theory
Euler's function, which is closely related to the Dedekind eta function, and occurs in the study of modular forms. The modulus of the Euler function (see
Pentagonal_number_theorem
Analytic function in mathematics
the Dirichlet L-functions and the Dedekind zeta function. For other related functions see the articles zeta function and L-function. The polylogarithm
Riemann_zeta_function
(2001), "Vassiliev invariants and a strange identity related to the Dedekind eta-function", Topology, 40 (5): 945–960, doi:10.1016/s0040-9383(00)00005-7,
Interval_order
Conjecture on zeros of the zeta function
extends the Riemann hypothesis to all Dedekind zeta functions of algebraic number fields. Since Dedekind zeta function for abelian extension of the rationals
Riemann_hypothesis
Unsolved problem in mathematics
(z)=q\prod _{n>0}\left(1-q^{n}\right)^{24}=\eta (z)^{24},} where η ( z ) {\textstyle \eta (z)} is the Dedekind eta function. Δ ( z ) {\textstyle \Delta (z)} is
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Algebra used in 2D conformal field theories and string theory
generating function for partitions, and is also written as q1/24 times the weight −1/2 modular form 1/η (the reciprocal of the Dedekind eta function). The
Vertex_operator_algebra
Special functions used to build correlation functions in 2D CFTs
_{P}(\tau )={\frac {q^{-P^{2}}}{\eta (\tau )}},} where η ( τ ) {\displaystyle \eta (\tau )} is the Dedekind eta function. Degenerate representation with
Virasoro_conformal_block
Conformal field theory of the 2D Ising model critical point
q ) {\displaystyle \eta (q)} is the Dedekind eta function, and θ i ( 0 | q ) {\displaystyle \theta _{i}(0|q)} are theta functions of the nome q = e 2
Two-dimensional critical Ising model
Two-dimensional_critical_Ising_model
functions G and H turn up in the Rogers–Ramanujan identities, and the function Q is the Euler function, which is closely related to the Dedekind eta function
Hard_hexagon_model
First article on transfinite set theory
Schwarz. Cantor's failure to mention Dedekind's contributions damaged his relationship with Dedekind. Dedekind stopped replying to his letters and did
Cantor's first set theory article
Cantor's_first_set_theory_article
Mathematical function
In mathematics, Hooley's delta function ( Δ ( n ) {\displaystyle \Delta (n)} ), also called Erdős--Hooley delta-function, defines the maximum number of
Hooley's_delta_function
Concept in abstract algebra
to the integers under addition. R {\displaystyle R} is a local ring, a Dedekind domain, and not a field. R {\displaystyle R} is Noetherian and a local
Discrete_valuation_ring
2D conformal field theories
q=e^{2\pi i\tau }} , and η ( τ ) {\displaystyle \eta (\tau )} is the Dedekind eta-function. This partition function is the sum of characters of the Virasoro algebra
Massless free scalar bosons in two dimensions
Massless_free_scalar_bosons_in_two_dimensions
Dutch-Australian number theorist
der Poorten, Alfred; Williams, Kenneth S. (1999), "Values of the Dedekind eta function at quadratic irrationalities", Canadian Journal of Mathematics,
Alfred_van_der_Poorten
spacetime 2. Dedekind eta function, a weight 1/2 modular form 3. Eta meson, a neutral flavor meson with PC = –+ θ 1. Theta function 2. θc is the Cabbibo
Glossary_of_string_theory
Subset of a preorder that contains all larger elements
{\displaystyle [5,\infty )} . In real analysis, a real number is often defined as a Dedekind cut. By definition, this is a nonempty proper lower subset of Q {\displaystyle
Upper_and_lower_sets
infinity; a Dedekind-finite set is a set that is not Dedekind-infinite. (These are also spelled without the hyphen, as "Dedekind finite" and "Dedekind infinite"
Glossary_of_set_theory
Function in number theory given by Srinivasa Ramanujan
new. (Papers, p. 179). In a footnote cites pp. 360–370 of the Dirichlet–Dedekind Vorlesungen über Zahlentheorie, 4th ed. Nathanson, ch. 8. Hardy & Wright
Ramanujan's_sum
Gives conditions for the solvability of quadratic equations modulo prime numbers
} Zeta function formulation As mentioned in the article on Dedekind zeta functions, quadratic reciprocity is equivalent to the zeta function of a quadratic
Quadratic_reciprocity
Algorithm for computing greatest common divisors
extended by Richard Dedekind, who used Euclid's algorithm to study algebraic integers, a new general type of number. For example, Dedekind was the first to
Euclidean_algorithm
Special mathematical function
was explored by Richard Dedekind and this function is the fundament in the theory of eta functions and their related functions. The elliptic nome is the
Nome_(mathematics)
Number, approximately 1.46557
value of Dedekind eta quotient ψ = e π i / 24 η ( τ ) 2 η ( 2 τ ) . {\displaystyle \psi ={\frac {e^{\pi i/24}\,\eta (\tau )}{{\sqrt {2}}\,\eta (2\tau )}}
Supergolden_ratio
Sets whose elements have degrees of membership
follows: A PFS A is characterized by three functions mapping U to [0, 1]: μ A , η A , ν A {\displaystyle \mu _{A},\eta _{A},\nu _{A}} , "degree of positive
Fuzzy_set
Characterizing property of mathematical constructions
Grothendieck group, completion of a metric space, completion of a ring, Dedekind–MacNeille completion, product topologies, Stone–Čech compactification,
Universal_property
System of mathematical set theory
η A ) . {\displaystyle \forall x(x\in a\iff x\;\eta \;A).} The statement a η C {\displaystyle a\;\eta \;C} where set a {\displaystyle a} represents class
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Number, approximately 1.3247
value of Dedekind eta quotient ρ = e π i / 24 η ( τ ) 2 η ( 2 τ ) . {\displaystyle \rho ={\frac {e^{\pi i/24}\,\eta (\tau )}{{\sqrt {2}}\,\eta (2\tau )}}
Plastic_ratio
the work of what is known as the "mathematical school", which included Dedekind, Pasch, Peano, Hilbert, Zermelo, Huntington, Veblen and Heyting. Their
History_of_logic
Number whose square is not the sum of 2 non-zero squares
Journal on Computing, 9 (1): 121–125, doi:10.1137/0209011, MR 0557832 OEIS sequence A004144 (Nonhypotenuse numbers) OEIS sequence A125667 (Eta numbers)
Nonhypotenuse_number
DEDEKIND ETA-FUNCTION
DEDEKIND ETA-FUNCTION
DEDEKIND ETA-FUNCTION
DEDEKIND ETA-FUNCTION
DEDEKIND ETA-FUNCTION
DEDEKIND ETA-FUNCTION
DEDEKIND ETA-FUNCTION
DEDEKIND ETA-FUNCTION
DEDEKIND ETA-FUNCTION