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Complex-valued arithmetic function
{\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} } is a Dirichlet character of modulus m {\displaystyle m} (where m {\displaystyle m} is a positive
Dirichlet_character
Type of mathematical function
Dirichlet character and s {\displaystyle s} a complex variable with real part greater than 1 {\displaystyle 1} . It is a special case of a Dirichlet series
Dirichlet_L-function
Sum in algebraic number theory
occur, for example, in the functional equations of Dirichlet L-functions, where for a Dirichlet character χ the equation relating L(s, χ) and L(1 − s, χ)
Gauss_sum
Type of character in number theory
Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L-functions larger than Dirichlet L-functions
Hecke_character
Meromorphic function on the complex plane
Dirichlet L-functions include the Riemann zeta function, since it arises from the trivial Dirichlet character modulo 1, i.e., the principal character
L-function
Mathematical construct
In mathematics, a character sum is a sum ∑ χ ( n ) {\textstyle \sum \chi (n)} of values of a Dirichlet character χ modulo N, taken over a given range of
Character_sum
Symbol in number theory
{\displaystyle \chi (n)=\left({\tfrac {a}{n}}\right)} is a real Dirichlet character of modulus { 4 | a | , a ≡ 2 ( mod 4 ) , | a | , otherwise. {\displaystyle
Kronecker_symbol
Mathematical conjecture about zeros of L-functions
Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis is
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
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
Conjecture on zeros of the zeta function
generalized RH is false for the L-function of some imaginary quadratic Dirichlet character then h(D) → ∞ as D → −∞. (In the work of Hecke and Heilbronn, the
Riemann_hypothesis
Mathematical function
are then called quasi-characters. Dirichlet characters can be seen as a special case of this definition. Multiplicative characters are linearly independent
Character_(mathematics)
Russian mathematician (1937–2008)
mathematician working in the field of analytic number theory, p-adic numbers and Dirichlet series. For most of his student and professional life he was associated
Anatoly_Karatsuba
Signed odd unit fractions sum to π/4
is the Dirichlet L-series of the non-principal Dirichlet character of modulus 4 evaluated at s = 1, and therefore the value β(1) of the Dirichlet beta function
Leibniz_formula_for_π
Topics referred to by the same term
Character theory, the mathematical theory of special kinds of characters associated to group representations Dirichlet character, a type of character
Character
Special character in number theory
numbers, ω {\displaystyle \omega } can be considered as a usual Dirichlet character of conductor q {\displaystyle q} . More generally, given a complete
Teichmüller_character
Special mathematical function
function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four. The Dirichlet beta function is defined as
Dirichlet_beta_function
Mathematical series
In mathematics, a Dirichlet series is any series of the form ∑ n = 1 ∞ a n n s , {\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}},} where s
Dirichlet_series
Potential counterexample to the generalized Riemann hypothesis
when the L-function is associated to a real Dirichlet character. For an integer q ≥ 1, a Dirichlet character modulo q is an arithmetic function χ : Z →
Siegel_zero
Formula in number theory
Then χ {\displaystyle \chi } is a Dirichlet character. Write L ( s , χ ) {\displaystyle L(s,\chi )} for the Dirichlet L-series based on χ {\displaystyle
Class_number_formula
Special functions of several complex variables
called the representation numbers of the form. For χ a primitive Dirichlet character modulo q and ν = 1 − χ(−1)/2 then θ χ ( z ) = 1 2 ∑ n = − ∞ ∞ χ
Theta_function
Function in number theory
reciprocity. Generalizations of the symbol include the Jacobi symbol and Dirichlet characters of higher order. The notational convenience of the Legendre symbol
Legendre_symbol
Twenty-second letter of the Greek alphabet
steel structures. In analytic number theory, chi is used for the Dirichlet character. U+03A7 Χ GREEK CAPITAL LETTER CHI (Χ) U+03C7 χ GREEK SMALL LETTER
Chi_(letter)
When a Gauss sum is the square root of a prime number, multiplied by a root of unity
says that for a Dirichlet character modulo an odd prime, if the argument of its Gaussian sum is a root of unity, then the character must be quadratic
Chowla–Mordell_theorem
Topics referred to by the same term
Conductor (ring theory) Conductor of an abelian variety Conductor of a Dirichlet character Conductor (class field theory) Artin conductor, of a Galois group
Conductor
Formal power series
_{p}(a_{n};p^{-s})\,.} If an is a Dirichlet character then its Dirichlet series generating function is called a Dirichlet L-series. We also have a relation
Generating_function
Topics referred to by the same term
arithmetic), base of modular arithmetic Similarly, the modulus of a Dirichlet character Moduli space, in mathematics a geometric space whose points represent
Modulus
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
theorem on sums of two squares) and χ {\displaystyle \chi } is the Dirichlet character from the Leibniz formula for π; also ∑ d | n χ ( d ) = ξ ( n ) {\displaystyle
Lemniscate_constant
Mathematical concept
of the Langlands dual group LG of G, generalizing the Dirichlet L-series of a Dirichlet character and the Mellin transform of a modular form. They were
Automorphic_L-function
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
Infinite products of functions indexed by primes
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product
Euler_product
Rational number sequence
{t^{k}}{k!}}.} Apart from the exceptional B1,1 = 1/2, we have, for any Dirichlet character χ, that Bk,χ = 0 if χ(−1) ≠ (−1)k. Generalizing the relation between
Bernoulli_number
Concept in mathematical group theory
_{1}(g)\chi _{2}(g)} . This group is connected to Dirichlet characters and Fourier analysis. The characters discussed in this section are assumed to be complex-valued
Character_theory
rings) Dirichlet algebra Dirichlet beta function Dirichlet boundary condition (differential equations) Neumann–Dirichlet method Dirichlet characters (number
List of things named after Peter Gustav Lejeune Dirichlet
List_of_things_named_after_Peter_Gustav_Lejeune_Dirichlet
Axiomatic definition of a class of L-functions
functional equation we would have Dirichlet L-functions for any imprimitive character. If χ {\textstyle \chi } is Dirichlet character induced by χ ⋆ {\textstyle
Selberg_class
Two-dimensional group theory table
This group is connected to Dirichlet characters and Fourier analysis. The outer automorphism group acts on the character table by permuting columns (conjugacy
Character_table
Arithmetic function
non-trivial example of a completely multiplicative function as are Dirichlet characters, the Jacobi symbol and the Legendre symbol. A completely multiplicative
Completely multiplicative function
Completely_multiplicative_function
{\displaystyle \chi } is a Dirichlet character modulo a cube free q ∈ N {\displaystyle q\in \mathbb {N} } that is not the principal character χ 0 {\displaystyle
Burgess_inequality
(s,\chi )=\varepsilon \Lambda (1-s,\chi ^{*})} with χ a primitive Dirichlet character, χ* its complex conjugate, Λ the L-function multiplied by a gamma-factor
Functional equation (L-function)
Functional_equation_(L-function)
Integer that is a perfect square modulo some integer
Vinogradov proved (independently) in 1918 that for any nonprincipal Dirichlet character χ(n) modulo q and any integers M and N, | ∑ n = M + 1 M + N χ ( n
Quadratic_residue
Symbols for constants, special functions
Fourier transform of a linear response function a character in mathematics; especially a Dirichlet character in number theory sometimes the mole fraction a
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
groups is in number theory, where it is used to construct Dirichlet characters. The character group of the cyclic group also appears in the theory of the
Character_group
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
Series representing modular forms
techniques involving holomorphic Eisenstein series twisted by a Dirichlet character produce formulas for the number of representations of a positive
Eisenstein_series
Number with an integer power equal to 1
Cyclotomic field Group scheme of roots of unity Dirichlet character Ramanujan's sum Witt vector Teichmüller character Hadlock, Charles R. (2000). Field Theory
Root_of_unity
Result on the class group of certain number fields, strengthening Ernst Kummer's theorem
in the range 1 to p − 1; we can therefore define a Dirichlet character ω (the Teichmüller character) with values in Z p {\displaystyle \mathbb {Z} _{p}}
Herbrand–Ribet_theorem
Identity obeyed by many special functions related to the gamma function
functions can be understood to be a special case, for the trivial Dirichlet character, of the Chowla–Selberg formula. Formally similar duplication formulas
Multiplication_theorem
Formula for the sum of an arithmetic function
A(x)=\sum _{n\leq x}\chi (n)} and χ ( n ) {\displaystyle \chi (n)} is a Dirichlet character. Other examples appear in the articles on the Mertens function and
Perron's_formula
Branch of number theory
functions are products of Dirichlet L-functions, with there being one factor for each Dirichlet character. The trivial character corresponds to the Riemann
Algebraic_number_theory
Anatolii A. Karatsuba and M. A. Korolev. Similarly, for any primitive Dirichlet character χ modulo q, we have N ( T , χ ) = T π log q T 2 π e + O ( log
Riemann–von_Mangoldt_formula
is taken over residue classes modulo p. More generally, given a Dirichlet character χ mod n, the Gauss sum mod n associated with χ is G ( k , χ ) = ∑
Gaussian_period
Generalization of the Riemann zeta function for algebraic number fields
used as a Dirichlet character. This fact—that the zeta function of a quadratic field is a product of the Riemann zeta function and the Dirichlet L-function
Dedekind_zeta_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
are then called quasi-characters. Dirichlet characters can be seen as a special case of this definition. Multiplicative characters are linearly independent
Multiplicative_character
Mathematical function
\left({\frac {\pi in^{2}\tau }{12}}\right),} where χ(n) is "the" Dirichlet character modulo 12 with χ(±1) = 1 and χ(±5) = −1. Explicitly,[citation needed]
Dedekind_eta_function
the only prime which is the sum of 4 consecutive primes. 24, all Dirichlet characters mod n are real if and only if n is a divisor of 24. 25, the first
List_of_numbers
Meissel–Mertens constant De Bruijn–Newman constant Dirichlet character Dirichlet L-series Siegel zero Dirichlet's theorem on arithmetic progressions Linnik's
List_of_number_theory_topics
Modular arithmetic concept
roots and quadratic residues. Artin's conjecture on primitive roots Dirichlet character Full reptend prime Multiplicative order Quadratic residue Root of
Primitive_root_modulo_n
(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
P-adic_L-function
Function whose domain is the positive integers
(n)=(-1)^{\Omega (n)}.} All Dirichlet characters χ(n) are completely multiplicative. Two characters have special notations: The principal character (mod n) is denoted
Arithmetic_function
Number-theoretic concept
sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J(χ, ψ) for Dirichlet characters χ, ψ modulo a prime
Jacobi_sum
Generalization of the Legendre symbol in number theory
permutation by Zolotarev's lemma. The Jacobi symbol (a/n) is a Dirichlet character to the modulus n. The above formulas lead to an efficient O(log a
Jacobi_symbol
Special function in mathematics
_{\chi }{\overline {\chi }}(n)L(s,\chi ),} the sum running over all Dirichlet characters mod k. In the opposite direction we have the linear combination L
Hurwitz_zeta_function
Function equal to the product of its values on coprime factors
( n ) {\displaystyle \tau (n)} : the Ramanujan tau function All Dirichlet characters are completely multiplicative functions, for example ( n / p ) {\displaystyle
Multiplicative_function
Concerns the class number of a real quadratic field of discriminant > 0
{\displaystyle m={\frac {d}{p}}\;} and χ {\displaystyle \chi \;} is the Dirichlet character for the quadratic field. For p = 3 there is a factor (1 + m) multiplying
Ankeny–Artin–Chowla congruence
Ankeny–Artin–Chowla_congruence
Russian mathematician (1921–1995)
number theory. He is known for the Postnikov character formula, which expresses the value of a Dirichlet character by means of a trigonometric function of
Alexey_Georgiyevich_Postnikov
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
Probabilistic primality test
even index, it suffices to assume the validity of GRH for quadratic Dirichlet characters. The running time of the algorithm is, in the soft-O notation, Õ((log
Miller–Rabin_primality_test
Mathematical concept
parameter. The Riemann zeta function can be replaced by a Dirichlet L-function of a Dirichlet character χ. The sum over prime powers then gets extra factors
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
Topics referred to by the same term
variable that exhibits a particular type of statistical independence (Dirichlet distribution) Neutrality (philosophy), the absence of declared or intentional
Neutral
Particular kind of exponential sum
Salié introduced a form of Kloosterman sum that is twisted by a Dirichlet character: Such Salié sums have an elementary evaluation. After the discovery
Kloosterman_sum
n^{2}z}\ ({\scriptstyle \nu ={\frac {1-\psi (-1)}{2}}})} for various Dirichlet characters ψ {\displaystyle \psi } then applies Weil's converse theorem. Theta
Shimura_correspondence
divisors of a natural number n {\displaystyle n} , or equivalently the Dirichlet convolution of an arithmetic function f ( n ) {\displaystyle f(n)} with
Divisor_sum_identities
Topological abelian group Haar measure Discrete Fourier transform Dirichlet character Amenable group Von Neumann's conjecture Pontryagin duality Kronecker's
List of harmonic analysis topics
List_of_harmonic_analysis_topics
mod 8. The required values of the Dirichlet series ζD(s) can be evaluated as follows. We write χ for the Dirichlet character with χ(m) given by 0 if m is even
Smith–Minkowski–Siegel mass formula
Smith–Minkowski–Siegel_mass_formula
\sum \chi (r)e(r/p)=G(\chi )} taken over r modulo p, where χ is a Dirichlet character taking values in the cube roots of unity, and where e(x) is the exponential
Kummer_sum
Theorem in analytic number theory
{\displaystyle \chi (n)} be a Dirichlet character modulo some natural number k ∈ N {\displaystyle k\in \mathbb {N} } . The associated Dirichlet L-function is defined
Grosswald–Schnitzer_theorem
Gaussian period Gaussian rational Gauss sum, an exponential sum over Dirichlet characters Elliptic Gauss sum, an analog of a Gauss sum Quadratic Gauss sum
List of things named after Carl Friedrich Gauss
List_of_things_named_after_Carl_Friedrich_Gauss
Math method
application of large sieves using estimations of mean values of Dirichlet characters. In the late 1960s and early 1970s, many of the key ingredients and
Large_sieve
Model from mathematical physics
representations take the role of integers, group characters taking the place the Dirichlet characters, and so on. D. J. G. Dueñas and N. F. Svaiter. Thermodynamics
Primon_gas
Topics referred to by the same term
individuals in the dataset Dirichlet process, a stochastic process corresponding to an infinite generalization of the Dirichlet distribution. Dynamic programming
DP
equation Diophantine equation Dirichlet 1. Dirichlet's theorem on arithmetic progressions 2. Dirichlet character 3. Dirichlet's unit theorem. distribution
Glossary_of_number_theory
Mathematical operation
transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and
Mellin_transform
Lejeune Dirichlet (1837) Pioneering paper in analytic number theory, which introduced Dirichlet characters and their L-functions to establish Dirichlet's theorem
List of publications in mathematics
List_of_publications_in_mathematics
Concept in number theory
Classical Dirichlet characters and ideal class characters occur as special cases. For example, over Q {\displaystyle \mathbb {Q} } , Dirichlet characters can
Idele_group
Russian mathematician (1792–1856)
known as Lobachevskian geometry, and also for his fundamental study on Dirichlet integrals, known as the Lobachevsky integral formula. William Kingdon
Nikolai_Lobachevsky
Type of theorem in automorphic forms
(n)a_{n}}{n^{s}}}} by some Dirichlet characters χ, satisfy suitable functional equations relating values at s and 1−s, then the Dirichlet series is essentially
Converse_theorem
}{\mathfrak {p}}}\right)_{n}} All power residue symbols mod n are Dirichlet characters mod n, and the m-th power residue symbol only contains the m-th roots
Power_residue_symbol
Concept in mathematics
also arises as the Euler-Lagrange equation of a functional called the Dirichlet energy. As such, the theory of harmonic maps contains both the theory
Harmonic_map
Complex-valued smooth functions of the upper half plane (harmonic analysis topic)
Folsom, Amanda (2014), "Almost harmonic Maass forms and Kac–Wakimoto characters", Journal für die Reine und Angewandte Mathematik, 2014 (694): 179–202
Maass_wave_form
Sum type in number theory
difficult: Gauss could only establish it after several years' work. Later, Dirichlet, Kronecker, Schur and other mathematicians found different proofs. Let
Quadratic_Gauss_sum
Describes statistically the splitting of primes in a given Galois extension of Q
be viewed as a generalisation of Dirichlet's theorem on arithmetic progressions. A quantitative form of Dirichlet's theorem states that if n ≥ 2 {\displaystyle
Chebotarev_density_theorem
Online database of integer sequences
because it comprehensively contains every OEIS field, filled. A046970 Dirichlet inverse of the Jordan function J_2 (A007434). 1, -3, -8, -3, -24, 24,
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Seventh letter in the Greek alphabet
designation. Mathematics, η-reduction in lambda calculus. Mathematics, the Dirichlet eta function, Dedekind eta function, and Weierstrass eta function. In
Eta
analytic number theory. In proving the theorem, he introduces the Dirichlet characters and L-functions. He also notes the difference between the absolute
1837_in_science
Generalized function whose value is zero everywhere except at zero
integrals only converge in the sense of distributions (an example is the Dirichlet kernel below), rather than in the sense of measures. Another example is
Dirac_delta_function
Divergent series
The latter series is an example of a Dirichlet series. When the real part of s is greater than 1, the Dirichlet series converges, and its sum is the Riemann
1_+_2_+_3_+_4_+_⋯
Type of generalization of periodic functions in Euclidean space
certain field extensions as Abelian groups. - Specific generalizations of Dirichlet L-functions as class field-theoretic objects. - Generally any harmonic
Automorphic_form
Analytic function on the upper half-plane with a certain behavior under the modular group
Number System". Quanta. Apostol, Tom M. (1990), Modular functions and Dirichlet Series in Number Theory, New York: Springer-Verlag, ISBN 0-387-97127-0
Modular_form
Number, approximately 3.14
higher-dimensional Poincaré inequalities that provide best constants for the Dirichlet energy of an n-dimensional membrane. Specifically, π is the greatest constant
Pi
thousands of years. Harald August Bohr (1952). Collected Mathematical Works: Dirichlet series. The Riemann Zeta-function. Dansk Matematisk Forening. p. xiv.
List_of_agnostics
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