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Meromorphic function on the complex plane
An L-function is a meromorphic function on the complex plane, and one out of several categories of mathematical objects studied in analytic number theory
L-function
Mathematical concept
mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
Type of mathematical function
In mathematics, a Dirichlet L-series is a function of the form L ( s , χ ) = ∑ n = 1 ∞ χ ( n ) n s , {\displaystyle L(s,\chi )=\sum _{n=1}^{\infty }{\frac
Dirichlet_L-function
Topics referred to by the same term
In mathematics, a Hecke L-function may refer to: an L-function of a modular form an L-function of a Hecke character This disambiguation page lists mathematics
Hecke_L-function
Mathematical concept
In mathematics, an automorphic L-function is a function L(s,π,r) of a complex variable s, associated to an automorphic representation π of a reductive
Automorphic_L-function
Type of Dirichlet series associated to number field extensions
In mathematics, Artin L-functions are a type of Dirichlet series defined for finite extensions of number fields, encoding informations about linear representations
Artin_L-function
Mathematical function associated to algebraic varieties
global L-function defined as an Euler product of local zeta functions. Hasse–Weil L-functions form one of the two major classes of global L-functions, alongside
Hasse–Weil_zeta_function
Analytic function in mathematics
p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose
P-adic_L-function
Conjecture on zeros of the zeta function
Unsolved problem in mathematics Do all non-trivial zeros of the Riemann zeta function have a real part equal to one half? More unsolved problems in mathematics
Riemann_hypothesis
Subfield of number theory
In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula
Special_values_of_L-functions
S-shaped curve
A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with the equation f ( x ) = L 1 + e − k ( x − x 0 ) {\displaystyle f(x)={\frac
Logistic_function
mathematics, motivic L-functions are a generalization of Hasse–Weil L-functions to general motives over global fields. The local L-factor at a finite place
Motivic_L-function
Analytic function in mathematics
of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known. The Riemann zeta function ζ ( s ) {\displaystyle
Riemann_zeta_function
Method of solution to differential equations
that if L {\displaystyle L} is a linear differential operator, then the Green's function G {\displaystyle G} is the solution of the equation L G = δ ,
Green's_function
Artin L-function is a function associated to a finite Galois extension of global fields created by packaging together the various Artin L-functions associated
Equivariant_L-function
Space of bounded sequences
Σ , μ ) {\displaystyle L^{\infty }=L^{\infty }(X,\Sigma ,\mu )} , the vector space of essentially bounded measurable functions with the essential supremum
L-infinity
Function studied by Ramanujan
of weight 12, it gives rise to an L {\displaystyle L} -function, called Ramanujan's L {\displaystyle L} -function. It is defined for R e ( s ) > 13 /
Ramanujan_tau_function
Methods used in statistics
variance stabilized Ripley K function called the L function is generally used. The sample version of the L function is defined as L ^ ( t ) = ( K ^ ( t ) π
Spatial descriptive statistics
Spatial_descriptive_statistics
Special mathematical function
is a particular Dirichlet L-function, the L-function for the alternating character of period four. The Dirichlet beta function is defined as β ( s ) = ∑
Dirichlet_beta_function
Point to which functions converge in analysis
f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More
Limit_of_a_function
Mathematical concept
In mathematics, the term standard L-function refers to a particular type of automorphic L-function described by Robert P. Langlands. Here, standard refers
Standard_L-function
Mathematical conjecture about zeros of L-functions
zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Mathematical function whose derivative exists
a real function f {\displaystyle f} , then f {\displaystyle f} is said to be differentiable at x 0 {\displaystyle x_{0}} if there exists an L ∈ R {\displaystyle
Differentiable_function
Concept in number theory (mathematics)
elliptic curve. The question of the rank is thought to be bound up with L-functions (see below). The torsor theory here leads to the Selmer group and Tate–Shafarevich
Arithmetic of abelian varieties
Arithmetic_of_abelian_varieties
Function whose squared absolute value has finite integral
square-integrable function, also called a quadratically integrable function or L 2 {\displaystyle L^{2}} function or square-summable function, is a real- or
Square-integrable_function
Unproved conjecture in mathematics
{\displaystyle K} and the behaviour of its associated Hasse–Weil L-function L ( E , s ) {\displaystyle L(E,s)} at s = 1 {\displaystyle s=1} . The points E ( K )
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Index of lists with the same name
function Ihara zeta function of a graph L-function, a "twisted" zeta function Lefschetz zeta function of a morphism Lerch zeta function, a generalization
List_of_zeta_functions
Type of character in number theory
to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which
Hecke_character
Unsolved problem in mathematics
plane, the Ramanujan L-function can be defined by analytic continuation of this series. Like other L-functions, the Ramanujan L-function satisfies a functional
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
researchers who discovered them. Functional equation Functional equation (L-function) Constitutive equation Laws of science Defining equation (physical chemistry)
List_of_equations
In mathematics, the Shimizu L-function, introduced by Hideo Shimizu in 1963, is a Dirichlet series associated to a totally real algebraic number field
Shimizu_L-function
Algebraic curve in mathematics
function of a complex variable, L, the Hasse–Weil zeta function of E over Q. This function is a variant of the Riemann zeta function and Dirichlet L-functions
Elliptic_curve
the coefficient of the leading term in the Taylor expansion of an Artin L-function associated with a Galois extension K/k of algebraic number fields. The
Stark_conjectures
Extension of the factorial function
Complex Function Theory. Translated by Kay, L. D. Springer. ISBN 978-0-387-98221-2. Lanczos, C. (1964). "A precision approximation of the gamma function". Journal
Gamma_function
In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional
Functional equation (L-function)
Functional_equation_(L-function)
Conjectures connecting number theory and geometry
L-function. One of his conjectures states that these L-functions satisfy a certain functional equation generalizing those of other known L-functions.
Langlands_program
In mathematics, a Shintani zeta function or Shintani L-function is a generalization of the Riemann zeta function. They were first studied by Takuro Shintani (1976)
Shintani_zeta_function
Belgian mathematician
important results on the l-adic representations attached to modular forms, and the conjectural functional equations of L-functions. Deligne also focused
Pierre_Deligne
Mathematic theory
locally compact group of ideles to lift the zeta function twisted by a Hecke character, i.e. a Hecke L-function, of a number field to a zeta integral and study
Tate's_thesis
Generalization of the Riemann zeta function for algebraic number fields
L / K ) {\displaystyle {\text{Gal}}(L/K)} , the resulting Artin L-function is: L ( s , 1 , L / K ) = ζ K ( s ) . {\displaystyle L(s,{\mathcal {1}},L/K)=\zeta
Dedekind_zeta_function
Zeta-like functions approximate arbitrary holomorphic functions
universality of zeta functions is the remarkable ability of the Riemann zeta function and other similar functions (such as the Dirichlet L-functions) to approximate
Zeta_function_universality
Special function in mathematics
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re(s) > 1 and a ≠ 0, −1
Hurwitz_zeta_function
Mathematical function
In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by Glaisher (1891). It is defined as the following infinite
Prime_zeta_function
Mathematical finite graph-associated function
mathematics, the Ihara zeta function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta function, and is used to relate
Ihara_zeta_function
Theorem in algebraic number theory relating p-adic L-functions and ideal class groups
main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa
Main conjecture of Iwasawa theory
Main_conjecture_of_Iwasawa_theory
Mathematical relation assigning a probability event to a cost
optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one
Loss_function
American mathematician (born 1945)
L-functions, which carry information for the unit root L-function studied by Dwork and Fu-Wan. Libgober and Sperber considered holomorphic functions from
Steven_Sperber
Mathematical Theory
representations of L-functions, is a technique for directly constructing and analytically continuing several important examples of automorphic L-functions. Some authors
Rankin–Selberg_method
Continuous function that is not absolutely continuous
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in
Cantor_function
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
}{\frac {\nu (n)}{n}}={\frac {\varpi }{4}}} where L {\displaystyle L} is the L-function of the elliptic curve E : y 2 = x 3 − x {\displaystyle E:\,y^{2}=x^{3}-x}
Lemniscate_constant
Ratio of polynomial functions
is L. The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K. A function f {\displaystyle
Rational_function
Function used in signal processing
processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside
Window_function
Mathematical function used in signal processing
processing, the function is sampled symmetrically (with spacing L / N {\displaystyle L/N} and amplitude 1 {\displaystyle 1} ): w [ n ] = L ⋅ w 0 ( L N ( n − N
Hann_function
Mathematical series
definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. Specifically, the Riemann zeta function ζ(s) is the Dirichlet
Dirichlet_series
expression appearing in the functional equation of an Artin L-function. Suppose that L {\displaystyle L} is a finite Galois extension of the local field K {\displaystyle
Artin_conductor
Summability method in physics
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent
Zeta_function_regularization
Transcendental single-variable function
tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function. The Clausen function of order 2 – often referred
Clausen_function
Mathematical description of quantum state
In quantum mechanics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common
Wave_function
Theorem on the number of primes in arithmetic sequences
Dirichlet (1837) with Dirichlet L-series. The proof is modeled on Euler's earlier work relating the Riemann zeta function to the distribution of primes
Dirichlet's theorem on arithmetic progressions
Dirichlet's_theorem_on_arithmetic_progressions
Theorem in analytic number theory
class of modified zeta functions and Dirichlet L-functions that possess exactly the same non-trivial zeros as the Riemann zeta function, but whose Euler products
Grosswald–Schnitzer_theorem
Mathematical concept in number theory
elliptic curve L-functions, including work related to the Birch and Swinnerton-Dyer conjecture and the Riemann hypothesis for such L-functions. In a branch
Partial_Euler_Product
Axiomatic definition of a class of L-functions
the essential properties satisfied by most functions that are commonly called L-functions or zeta functions. Although the exact nature of the class is
Selberg_class
Complex-valued arithmetic function
theory and related branches of mathematics, a complex-valued arithmetic function χ : Z → C {\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} } is
Dirichlet_character
L-functions. Weil returned to this idea in a 1972 paper, showing how the formulation extended to a larger class of L-functions (Artin-Hecke L-functions);
Weil's_criterion
K/k. The S-imprimitive equivariant Artin L-function θ(s) is obtained from the usual equivariant Artin L-function by removing the Euler factors corresponding
Brumer–Stark_conjecture
Function related to statistics and probability theory
variable with probability mass function p {\textstyle p} depending on a parameter θ {\textstyle \theta } . Then the function L ( θ ∣ x ) = p θ ( x ) = P θ
Likelihood_function
Mathematical theorem
theorems of global class field theory. It can be used to prove that Artin L-functions are meromorphic, and also to prove the Chebotarev density theorem. Two
Artin_reciprocity
Constants of the mathematical zeta function
In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted ζ ( s ) {\displaystyle
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
German mathematician (born 1958)
Deninger's research focuses on arithmetic geometry, including applications to L-functions. Deninger obtained his doctorate from the University of Cologne in 1982
Christopher_Deninger
Arithmetic function related to the divisors of an integer
theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number
Divisor_function
Sigmoid shape special function
mathematics, the error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ( z ) = 2
Error_function
Function specifying the behavior of a component in an electronic or control system
a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that models
Transfer_function
mathematics, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as Z ( V , s ) =
Local_zeta_function
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
Potential counterexample to the generalized Riemann hypothesis
theory of Dirichlet L-functions is as potential exceptions to the classical zero-free regions, which can only occur when the L-function is associated to
Siegel_zero
Multivalued function in mathematics
) 6 L 1 3 + L 2 ( − 12 + 36 L 2 − 22 L 2 2 + 3 L 2 3 ) 12 L 1 4 + ⋯ = L 1 − L 2 + ∑ ℓ = 1 ∞ 1 L 1 ℓ ∑ m = 1 ℓ ( − 1 ) ℓ − m [ ℓ ℓ − m + 1 ] m ! L 2 m
Lambert_W_function
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
the 1960s meant that Hasse–Weil L-functions could be regarded as Artin L-functions for the Galois representations on l-adic cohomology groups. Bad reduction
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Topics referred to by the same term
meromorphic function on the complex plane L {\displaystyle {\mathcal {L}}} , Laplace transform L {\displaystyle {\mathcal {L}}} , likelihood function ℓp space
L_(disambiguation)
Topics referred to by the same term
L series may refer to: L-function, a meromorphic function Dirichlet L-function, in number theory Artin L-function, a type of Dirichlet series Canon L
L_series
the Langlands–Shahidi method provides the means to define automorphic L-functions in many cases that arise with connected reductive groups over a number
Langlands–Shahidi_method
Mathematical function having a characteristic S-shaped curve or sigmoid curve
sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the
Sigmoid_function
Warren at MIT. The Patterson function is defined as P ( u , v , w ) = ∑ h , k , ℓ ∈ Z | F h , k , ℓ | 2 e − 2 π i ( h u + k v + ℓ w ) . {\displaystyle P(u
Patterson_function
Type of polynomial
< t < 1 implies an absence of the same kind for the L-function L ( s , x p ) . {\displaystyle L\left(s,{\dfrac {x}{p}}\right).\,} This is of considerable
Fekete_polynomial
Exploring properties of the integers with complex analysis
with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions
Analytic_number_theory
Association of one output to each input
mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the
Function_(mathematics)
On generating functions from counting points on algebraic varieties over finite fields
I − TF on the ℓ-adic cohomology group Hi. The rationality of the zeta function follows immediately. The functional equation for the zeta function follows from
Weil_conjectures
conjecture on primitive roots Artin conjecture on L-functions Artin group Artin–Hasse exponential Artin L-function Artin reciprocity Artin–Rees lemma Artin representation
List of things named after Emil Artin
List_of_things_named_after_Emil_Artin
Simpler variant of the Riemann zeta function
Riemann xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named
Riemann_xi_function
Complex-valued smooth functions of the upper half plane (harmonic analysis topic)
smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup Γ {\displaystyle \Gamma } of S L 2 (
Maass_wave_form
Elliott–Halberstam conjecture Functional equation (L-function) Chebotarev's density theorem Local zeta function Weil conjectures Modular form modular group Congruence
List_of_number_theory_topics
In mathematics, the Brownian motion and the Riemann zeta function are two central objects of study in mathematics originating from different fields - probability
Brownian motion and Riemann zeta function
Brownian_motion_and_Riemann_zeta_function
German algebraic number theorist
algebraic number theorist at ETH Zurich. Her research interests include L-functions, modular forms, p-adic Hodge theory, and Iwasawa theory, and her work
Sarah_Zerbes
The Selberg zeta-function was introduced by Atle Selberg (1956). It is analogous to the famous Riemann zeta function ζ ( s ) = ∏ p ∈ P 1 1 − p − s {\displaystyle
Selberg_zeta_function
Conjecture in the representation theory of Lie groups
_{n}\boxtimes \mathrm {std} _{n-1})} where L E {\displaystyle L_{E}} is the global L-function obtained as the product of local L-factors given by the local Langlands
Gan–Gross–Prasad_conjecture
Topics referred to by the same term
be assigned In number theory, the value of an L-function In space physics, the value assigned to an L-shell, a particular set of planetary magnetic field
L-value
Dirichlet beta function Dirichlet L-function Hurwitz zeta function Legendre chi function Lerch transcendent Polylogarithm and related functions: Incomplete
List of mathematical functions
List_of_mathematical_functions
_{n=1}^{\infty }{2^{n}P(2n) \over n}\right)\right).} Riemann zeta function L-function Euler product Twin prime "Feller–Tornier Constant – from Wolfram
Feller–Tornier_constant
Chemical compound
magnesium ions. One small study reports that magnesium L-threonate improves cognitive function in healthy adults, compared to placebo; other magnesium
Magnesium_L-threonate
Generalizations of the Riemann zeta function
In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by ζ ( s 1 , … , s k ) = ∑ n 1 > n 2 > ⋯ > n k >
Multiple_zeta_function
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