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Indicator function of rational numbers
In mathematics, the Dirichlet function is the indicator function 1 Q {\displaystyle \mathbf {1} _{\mathbb {Q} }} of the set of rational numbers Q {\displaystyle
Dirichlet_function
Type of mathematical function
In mathematics, a Dirichlet L-series is a function of the form L(s,χ)=∑n=1∞χ(n)ns, where χ is a Dirichlet character and s a complex variable with real
Dirichlet_L-function
Function in analytic number theory
in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number
Dirichlet_eta_function
Special mathematical function
mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a
Dirichlet_beta_function
Meromorphic function on the complex plane
hypothesis and its generalisations. A Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation, is
L-function
Mathematical series
Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. Specifically, the Riemann zeta function ζ(s) is the Dirichlet series of the
Dirichlet_series
Function that is discontinuous at rationals and continuous at irrationals
names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function (not to be confused
Thomae's_function
Function which is not continuous at any point of its domain
indicator function of the rational numbers, also known as the Dirichlet function. This function is denoted as 1 Q {\displaystyle \mathbf {1} _{\mathbb {Q}
Nowhere_continuous_function
German mathematician (1805–1859)
Johann Peter Gustav Lejeune Dirichlet (/ˌdɪərɪˈkleɪ/; German: [ləˈʒœn diʁiˈkleː]; 13 February 1805 – 5 May 1859) was a German mathematician. In number
Peter Gustav Lejeune Dirichlet
Peter_Gustav_Lejeune_Dirichlet
Formal power series
generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every
Generating_function
Mathematical operation on arithmetical functions
In mathematics, Dirichlet convolution (or divisor convolution) is a binary operation defined for arithmetic functions; it is important in number theory
Dirichlet_convolution
Function studied by Ramanujan
Sequences. 13: Article 10.7.4. Apostol, Tom M. (1990) [1976]. Modular Functions and Dirichlet Series in Number Theory. Graduate Texts in Mathematics. Vol. 41
Ramanujan_tau_function
Type of constraint on solutions to differential equations
weight function w appearing in the boundary expression is termed a primary variable, and its specification constitutes the essential or Dirichlet boundary
Dirichlet_boundary_condition
Integral of sin(x)/x from 0 to infinity
the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over
Dirichlet_integral
Infinite series in mathematical analysis
In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of ∑n=1∞ane−λns, where an, s are complex numbers
General_Dirichlet_series
and rings) Dirichlet algebra Dirichlet beta function Dirichlet boundary condition (differential equations) Neumann–Dirichlet method Dirichlet characters
List of things named after Peter Gustav Lejeune Dirichlet
List_of_things_named_after_Peter_Gustav_Lejeune_Dirichlet
Mathematical function characterizing set membership
belonging to A; that is, 𝟏A(x)=[ x∈A ]. For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers
Indicator_function
Complex-valued arithmetic function
a complex-valued arithmetic function χ : Z → C {\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} } is a Dirichlet character of modulus m {\displaystyle
Dirichlet_character
Probability distribution
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted Dir ( α ) {\displaystyle \operatorname
Dirichlet_distribution
Synchrotron function Riemann zeta function: A special case of Dirichlet series. Riemann Xi function Dirichlet eta function: An allied function. Dirichlet beta
List of mathematical functions
List_of_mathematical_functions
Function with a repeating pattern
periodic but possess properties that make them less intuitive. The Dirichlet function, for example, is periodic, with any nonzero rational number serving
Periodic_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
Point to which functions converge in analysis
x-coordinate. The function f(x)={1x rational 0x irrational (a.k.a., the Dirichlet function) has no limit at any x-coordinate. The function f(x)={1 for x<02 for x≥0
Limit_of_a_function
Method of mathematical integration
continuous functions, including elementary functions, for example polynomials. However, the graphs of other functions, for example the Dirichlet function, do
Lebesgue_integral
the characteristic function of the rational numbers, χ Q {\displaystyle \chi _{\mathbb {Q} }} , also known as the Dirichlet function which is discontinuous
Baire_function
Modern application of infinitesimals
extension of the Dirichlet function takes different values (0 and 1) at these two infinitely close points, and therefore the Dirichlet function is not continuous
Nonstandard_calculus
Problem of solving a partial differential equation subject to prescribed boundary values
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region
Dirichlet_problem
Functions such that f(–x) equals f(x) or –f(x)
multiplication. A function's being odd or even does not imply differentiability, or continuity. For example, the Dirichlet function is even, but is nowhere
Even_and_odd_functions
Operation in differential calculus
derivative is finite at 0, i.e. this is an essential discontinuity. The Dirichlet function, defined as: f ( x ) = { 1 , if x is rational 0 , if x is irrational
Symmetric_derivative
Concept in mathematical analysis
In mathematical analysis, the Dirichlet kernel, is the collection of periodic functions defined as D n ( x ) = ∑ k = − n n e i k x = 1 + 2 ∑ k = 1 n cos
Dirichlet_kernel
Generalized function whose value is zero everywhere except at zero
Fourier series states that the Dirichlet kernel restricted to the interval [−π,π] tends to a multiple of the delta function as N → ∞. This is interpreted
Dirac_delta_function
Mathematical measure of a function's variability
the Dirichlet energy is a measure of how variable a function is. More abstractly, it is a quadratic functional on the Sobolev space H1. The Dirichlet energy
Dirichlet_energy
Functions in mathematics
is Dirichlet's principle, representing harmonic functions in the Sobolev space H 1 ( {\displaystyle H^{1}(} as the minimizers of the Dirichlet energy
Harmonic_function
Second-order partial differential equation
solution to the corresponding Dirichlet problem. The Neumann boundary conditions for Laplace's equation specify not the function φ itself on the boundary of
Laplace's_equation
Method of solution to differential equations
the electric field. If the problem is to solve a Dirichlet boundary value problem, the Green's function should be chosen such that G(x,x′) vanishes when
Green's_function
Multiplicative function in number theory
function are plotted below: Larger values can be checked in: Wolframalpha the b-file of OEIS The Dirichlet series that generates the Möbius function is
Möbius_function
Theorem on the number of primes in arithmetic sequences
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there
Dirichlet's theorem on arithmetic progressions
Dirichlet's_theorem_on_arithmetic_progressions
Arithmetic function
all over the prime numbers. Arithmetic function Dirichlet L-function Dirichlet series Multiplicative function Apostol (1976), p. 30 Apostol (1976), p
Completely multiplicative function
Completely_multiplicative_function
Mathematical conjecture about zeros of L-functions
are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Theorem in combinatorics
function whose codomain is smaller than its domain". Advanced mathematical proofs like Siegel's lemma build upon this more general concept. Dirichlet
Pigeonhole_principle
Theorem
In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a complex-valued, periodic function f {\displaystyle f} to be equal to the sum
Dirichlet–Jordan_test
Mathematical function whose set of values is bounded
number (cf. Dirichlet function) is bounded. Thus, a function does not need to be "nice" in order to be bounded. The set of all bounded functions defined on
Bounded_function
Function that attains finitely many values
is the Dirichlet function over the real line, which takes the value 1 if x is rational and 0 otherwise. (Thus the "simple" of "simple function" has a
Simple_function
Concept in potential theory
least one function making the Dirichlet's integral finite). This concept is named after the German mathematician Peter Gustav Lejeune Dirichlet. The name
Dirichlet's_principle
Special mathematical function defined as sin(x)/x
integrals Dirichlet integral – Integral of sin(x)/x from 0 to infinity Lanczos resampling – Technique in signal processing List of mathematical functions Shannon
Sinc_function
Dunkl–Cherednik operator Dickman–de Bruijn function Peter Gustav Lejeune Dirichlet: Dirichlet function, Dirichlet L-function Engel: Engel expansion Erdélyi Artúr:
List of eponyms of special functions
List_of_eponyms_of_special_functions
Any real function on R admits a continuous restriction on a dense subset of R
Henry Blumberg. For instance, the restriction of the Dirichlet function (the indicator function of the rational numbers Q {\displaystyle \mathbb {Q} }
Blumberg_theorem
Counterintuitive mathematical object
example, the Dirichlet function is Lebesgue integrable, and convolution with test functions is used to approximate any locally integrable function by smooth
Pathological_(mathematics)
Topics referred to by the same term
In mathematics, 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
Function whose domain is the positive integers
μ(n), the Möbius function, is important because of the Möbius inversion formula. See § Dirichlet convolution, below.
Arithmetic_function
example, the Dirichlet function. Locally constant function: a continuous function into a discrete space. Homeomorphism: is a bijective function that is also
List_of_types_of_functions
On converting relations to functions of several real variables
In multivariable calculus, the implicit function theorem is a theorem that provides sufficient conditions under which a planar curve specified by F ( x
Implicit_function_theorem
Function equal to the product of its values on coprime factors
{\displaystyle \tau (n)} : the Ramanujan tau function All Dirichlet characters are completely multiplicative functions, for example ( n / p ) {\displaystyle
Multiplicative_function
Class of mathematical functions
1017/cbo9780511791246. ISBN 978-0-521-53429-1. Apostol, Tom M. (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag
Weierstrass_elliptic_function
Number of integers coprime to and less than n
proof of Dirichlet's theorem on arithmetic progressions. The Dirichlet series for φ(n) may be written in terms of the Riemann zeta function as: ∑ n =
Euler's_totient_function
Mathematical operation
theory, a Dirichlet series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic functions in a meaningful
Dirichlet_series_inversion
Formula for the derivative of an inverse function
calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the
Inverse_function_rule
Family of stochastic processes
In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes
Dirichlet_process
Theorem in mathematics
mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that if
Inverse_function_theorem
Inputs for which a function's value is non-zero
smaller than the closed support. For example, if f:[0,1]→ℝ is the Dirichlet function that is 0 on irrational numbers and 1 on rational numbers, and [0
Support_(mathematics)
Mathematical relation consisting of a multi-variable function equal to zero
multivariable functions that are continuously differentiable. A common type of implicit function is an inverse function. Not all functions have a unique
Implicit_function
Conjecture on zeros of the zeta function
this continuation observes that the series for the zeta function and the Dirichlet eta function satisfy the relation (1−22s)ζ(s)=η(s)=∑n=1∞(−1)n+1ns=11s−12s+13s−⋯
Riemann_hypothesis
integer-valued function. Integer-valued functions defined on the domain of all real numbers include the floor and ceiling functions, the Dirichlet function, the
Integer-valued_function
Smooth approximation of one-hot arg max
Multinomial logistic regression Dirichlet distribution – an alternative way to sample categorical distributions Partition function Exponential tilting – a generalization
Softmax_function
Type of character in number theory
generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting
Hecke_character
Generative topic model
In natural language processing, latent Dirichlet allocation (LDA) is a generative statistical model that explains how a collection of text documents can
Latent_Dirichlet_allocation
Generalization of the Riemann zeta function for algebraic number fields
of the Riemann zeta function: it can be defined as a Dirichlet series; it has an analytic continuation to a meromorphic function on the complex plane
Dedekind_zeta_function
About mathematical functions
definition that became known as Dirichlet's definition." Edwards also credits Euler with a general concept of a function and says further that The relations
History of the function concept
History_of_the_function_concept
Summatory function of the divisor-counting function
the Dirichlet divisor problem". Acta Arithmetica. 60 (4): 389–415. doi:10.4064/aa-60-4-389-415. ISSN 0065-1036. S2CID 59450869. Theorem 1 The function has
Divisor_summatory_function
Notion in calculus
calculus, the differential represents the principal part of the change in a function y = f ( x ) {\displaystyle y=f(x)} with respect to changes in the independent
Differential_of_a_function
Mathematical function with no sudden changes
ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854. A real function (that is, a function from real numbers to real numbers) can be represented
Continuous_function
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
Differential operator in mathematics
in physics is that solutions to Δf = 0 in a region U are functions that make the Dirichlet energy functional stationary: E ( f ) = 1 2 ∫ U ‖ ∇ f ‖ 2
Laplace_operator
Summability method in physics
series Minakshisundaram–Pleijel zeta function Zeta function (operator) ^ Tom M. Apostol, "Modular Functions and Dirichlet Series in Number Theory", "Springer-Verlag
Zeta_function_regularization
Topics referred to by the same term
Look up lambda function in Wiktionary, the free dictionary. Lambda function may refer to: Dirichlet lambda function, λ(s) = (1 – 2−s)ζ(s) where ζ is the
Lambda_function
Exploring properties of the integers with complex analysis
begun 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
Matrix of second derivatives
partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix
Hessian_matrix
Probability distribution
Dirichlet distribution, the solution can be written in terms of the digamma ψ {\displaystyle \psi } and trigamma ψ ′ {\displaystyle \psi '} functions:
Logit-normal_distribution
Conditions for switching order of integration in calculus
{\pi }{2}}\ln(2)\end{aligned}}} The Dirichlet series defines the Dirichlet eta function as follows: η ( s ) = ∑ n = 1 ∞ ( − 1 ) n − 1 n s
Fubini's_theorem
Modes of vibration in mathematics
(1) is often known as the Dirichlet Laplacian when it is considered as accepting only functions u satisfying the Dirichlet boundary condition. More generally
Dirichlet_eigenvalue
Potential counterexample to the generalized Riemann hypothesis
counterexample to the generalized Riemann hypothesis, on the zeros of Dirichlet L-functions associated to quadratic number fields. Roughly speaking, these would
Siegel_zero
Multivariate derivative (mathematics)
gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) ∇f whose value at a point p gives the
Gradient
Theorem in measure theory
first n rationals and 0 otherwise. Then f {\displaystyle f} is the Dirichlet function on [ 0 , 1 ] {\displaystyle [0,1]} , which is not Riemann integrable
Dominated_convergence_theorem
Analytic function in mathematics
(a)te^{at}}{e^{ft}-1}}} for χ a Dirichlet character with conductor f. The Kubota–Leopoldt p-adic L-function Lp(s, χ) interpolates the Dirichlet L-function with the Euler
P-adic_L-function
Function on an integer n which is log(p) if n equals p^k and zero otherwise
The von Mangoldt function plays an important role in the theory of Dirichlet series, and in particular, the Riemann zeta function. For example, one has
Von_Mangoldt_function
Theorem in measure theory
apparent, as can be demonstrated by example. Consider Dirichlet function, that is the indicator function 1 Q : [ 0 , 1 ] → { 0 , 1 } {\displaystyle 1_{\mathbb
Lusin's_theorem
Instantaneous rate of change (mathematics)
quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input
Derivative
Type of plane partition
Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi cells are also known as Thiessen polygons
Voronoi_diagram
Mathematical form
harmonic functions) and functional analysis, Dirichlet forms generalize the Laplacian (the mathematical operator on scalar fields). Dirichlet forms can
Dirichlet_form
terms of the zeta function. The function δ {\displaystyle \delta } is multiplicative, and since it is bounded by 1, its Dirichlet series converges absolutely
Average order of an arithmetic function
Average_order_of_an_arithmetic_function
Concept in number theory
In mathematics, the Dirichlet density (or analytic density) of a set of primes, named after Peter Gustav Lejeune Dirichlet, is a measure of the size of
Dirichlet_density
Mathematical concept
Here y is a real parameter. The Riemann zeta function can be replaced by a Dirichlet L-function of a Dirichlet character χ. The sum over prime powers then
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
Mathematical approximation of a function
of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the
Taylor_series
Explicit formula (L-function) Riemann–Siegel formula (particular approximate functional equation) "§25.15 Dirichlet -functions on NIST". Weisstein, Eric
Functional equation (L-function)
Functional_equation_(L-function)
Test for series convergence
In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence
Dirichlet's_test
Special function in mathematics
the Hurwitz zeta function may be expressed as a linear combination of Dirichlet L-functions and vice versa: The Hurwitz zeta function coincides with Riemann's
Hurwitz_zeta_function
Arithmetic function related to the divisors of an integer
is known as Dirichlet's divisor problem. The behaviour of the sigma function is irregular. The asymptotic growth rate of the sigma function can be expressed
Divisor_function
Index of lists with the same name
function, the special case for the poset of integer divisibility is related as a formal Dirichlet series to the Riemann zeta function. Zeta function of
List_of_zeta_functions
Arithmetical function
0\ldots } is the characteristic function of the squares, another Dirichlet convolution leads to the generalized σ-function, ϵ 2 ( n ) ∗ ψ k ( n ) = σ k (
Dedekind_psi_function
Special functions of several complex variables
mathematics, theta functions are special functions of several complex variables. Fundamentally, they are a family of continuous functions which encode the
Theta_function
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DIRICHLET FUNCTION
DIRICHLET FUNCTION
Biblical
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Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Male
Egyptian
, an Egyptian functionary.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Male
Celtic
, great justiciary, or functionary.
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Male
Egyptian
, Functionary of the Interior.
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, the son of the functionary Heknofre.
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Male
Egyptian
, a great functionary.
Male
Egyptian
, a high Egyptian functionary.
DIRICHLET FUNCTION
DIRICHLET FUNCTION
DIRICHLET FUNCTION
DIRICHLET FUNCTION
DIRICHLET FUNCTION
DIRICHLET FUNCTION
DIRICHLET FUNCTION
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