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Approximate identity involving logarithms of primes
In number theory, Selberg's identity is an approximate identity involving logarithms of primes found by Atle Selberg. The identity forms the crucial starting
Selberg's_identity
Norwegian mathematician (1917–2007)
Selberg archive webpage Obituary at Institute for Advanced Study Obituary in The Times Atle Selbergs private archive exists at NTNU University Library
Atle_Selberg
Mathematical theorem
In mathematics, the Selberg trace formula, introduced by Selberg (1956), is an expression for the character of the unitary representation of a Lie group
Selberg_trace_formula
Conjecture on zeros of the zeta function
The Selberg trace formula is the analogue for these functions of the explicit formulas in prime number theory. Selberg proved that the Selberg zeta functions
Riemann_hypothesis
Mathematical function
+\beta +(n+j-1)\gamma )\Gamma (1+\gamma )}}\end{aligned}}} Selberg's formula implies Dixon's identity for well poised hypergeometric series, and some special
Selberg_integral
On finite sums of products of three binomial coefficients, and a hypergeometric sum
mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon
Dixon's_identity
Particular kind of exponential sum
-1}}+\zeta _{2^{\alpha -1}}^{-1}\right)} for 2α || m with α > 3. The Selberg identity: K ( a , b ; m ) = ∑ d ∣ gcd ( a , b , m ) d ⋅ K ( a b d 2 , 1 ; m
Kloosterman_sum
Mathematical concept
central idea of the method is expressed by the following identity, sometimes called the Legendre identity: S ( A , P ) = ∑ a ∈ A ∑ d ∣ a ; d ∣ P μ ( d ) = ∑
Legendre_sieve
Theorem in Lie theory in mathematics
the identity element such that every lattice in the group has a conjugate whose intersection with this neighbourhood contains only the identity. This
Kazhdan–Margulis_theorem
Equation in Fourier analysis
non-commutative harmonic analysis, the idea is taken even further in the Selberg trace formula but takes on a much deeper character. A series of mathematicians
Poisson_summation_formula
Function equal to the product of its values on coprime factors
{\displaystyle 1(n)=1} Id ( n ) {\displaystyle \operatorname {Id} (n)} : the identity function, defined by Id ( n ) = n {\displaystyle \operatorname {Id} (n)=n}
Multiplicative_function
Mathematical function
is available at the Wayback machine of Michael Somos' website. Chowla–Selberg formula Ramanujan–Sato series q-series Weierstrass elliptic function Partition
Dedekind_eta_function
Identity obeyed by many special functions related to the gamma function
certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product
Multiplication_theorem
Israeli mathematician
proved the Macdonald-Selberg conjecture for the infinite Lie algebras of type B, C, and D. Regev, Amitai (1972). "Existence of identities in A⊗B". Israel Journal
Amitai_Regev
Math method
classes of numbers are removed, as opposed to small sieves such as the Selberg sieve wherein only a few residue classes are removed. The method has been
Large_sieve
(pseudo-)Riemannian manifold whose geodesics are reversible
be the identity involution (indicated by a dash). In the above tables this is implicitly covered by the case kl = 0. In the 1950s Atle Selberg extended
Symmetric_space
Online information retrieval tool
incorporate the idea of meta searching was University of Washington student Eric Selberg, who published a paper about his MetaCrawler experiment in 1995. The search
Metasearch_engine
Country in northern Europe
logic, while Øystein Ore and Ludwig Sylow advanced group theory. Atle Selberg, a major figure in 20th-century mathematics, was honoured with the Fields
Norway
Ways to estimate the size of sifted sets of integers
{\displaystyle E_{1}:=A} . We can rewrite the sifting function with Legendre's identity S ( A , P , z ) = ∑ d ∣ P ( z ) μ ( d ) A d ( x ) {\displaystyle S({\mathcal
Sieve_theory
Meromorphic function on the complex plane
today referred to as the Selberg class. The overarching hypothesis and the motivating background for the definition of the Selberg class is the so-called
L-function
polynomial, Rogers–Ramanujan identity, Rogers–Szegő polynomials Schubert polynomial Issai Schur: Schur polynomial Atle Selberg: Selberg integral Sheffer polynomial
List of eponyms of special functions
List_of_eponyms_of_special_functions
Decomposition of an integer as a sum of positive integers
6.159. Zbl 0071.04004. (Provides the Selberg formula. The older form is the finite Fourier expansion of Selberg.) "Partition", Encyclopedia of Mathematics
Integer_partition
{\displaystyle t} ? Generalized Riemann hypothesis for Selberg class: do the nontrivial zeros of all functions in Selberg class lie on the critical line 1 / 2 + i t
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Theorem about the constant term of certain Laurent polynomials
d\theta _{1}\cdots d\theta _{n}.} Dyson's integral is a special case of Selberg's integral after a change of variable and has value Γ ( 1 + β n / 2 ) Γ
Dyson_conjecture
Mathematical constants
x = 1 and x ≈ 3.5623822853908976914156443427... OEIS: A218802. Chowla–Selberg formula Waldschmidt, Michel (2006). "Transcendence of periods: the state
Particular values of the gamma function
Particular_values_of_the_gamma_function
Least common multiple Euclidean algorithm Coprime Euclid's lemma Bézout's identity, Bézout's lemma Extended Euclidean algorithm Table of divisors Prime number
List_of_number_theory_topics
Analytic function in mathematics
zeta function and prime numbers was discovered by Euler, who proved the identity ∑ n = 1 ∞ 1 n s = ∏ p prime 1 1 − p − s , {\displaystyle \sum _{n=1}^{\infty
Riemann_zeta_function
Theorem in abstract algebra
strategy for proving local and global Langlands conjectures using the Arthur–Selberg trace formula, but in order for this approach to work, the geometric sides
Fundamental lemma (Langlands program)
Fundamental_lemma_(Langlands_program)
Characterization of how many integers are prime
of the Erdős–Selberg proof of the PNT. This was the first machine-verified proof of the PNT. Avigad chose to formalize the Erdős–Selberg proof rather
Prime_number_theorem
Norwegian mathematician (1802–1829)
1899 to complement the Nobel Prizes, it was first awarded in 2003, while Selberg received an honorary Abel Prize the previous year. Mathematician Felix
Niels_Henrik_Abel
Special function defined by an integral
(First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 298 Chowla, S.; Selberg, A. (1949). "On Epstein's Zeta Function (I)". Proceedings of the National
Elliptic_integral
Arithmetic function related to the divisors of an integer
(including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein
Divisor_function
Japanese mathematician and professor
contributions to the theory of determinants are the extension of the Capelli identity to the context of quantum group G L q ( n ) {\displaystyle GL_{q}(n)}
Masato_Wakayama
British theoretical physicist and mathematician (1923–2020)
paper that inspired John Ward to derive his celebrated Ward–Takahashi identity. Dyson joined the faculty at Cornell as a physics professor in 1951, though
Freeman_Dyson
American mathematician (1942–2012)
Siegel–Weil formula: the regularized Siegel–Weil formula and the first term identity. These results have prompted other mathematicians to extend Siegel–Weil
Stephen_Rallis
Series whose partial sums eventually only have a fixed number of terms after cancellation
telescopic canceling between the consecutive terms. Using the angle addition identity for a product of sines, ∑ n = 1 N sin ( n ) = ∑ n = 1 N 1 2 csc ( 1
Telescoping_series
Symmetric holomorphic function
gamma function for any x ∈ Q + {\displaystyle x\in \mathbb {Q} ^{+}} , as Selberg and Chowla proved in 1949. The following expression is valid for all n
Modular_lambda_function
Proof that only uses basic techniques
of Copenhagen. Quoted in Goldfeld (2003), p. 3 However, in 1948, Atle Selberg produced new methods which led him and Paul Erdős to find elementary proofs
Elementary_proof
Branch of pure mathematics
it was constructed by means of what amounts, in modern language, to the identity ( 1 2 ( x − 1 x ) ) 2 + 1 = ( 1 2 ( x + 1 x ) ) 2 , {\displaystyle \left({\frac
Number_theory
American ventriloquist and actor (1922–2005)
wife in an accident). In 1996, Winchell contracted with figure maker Tim Selberg to construct a more contemporary version of Jerry Mahoney, which Winchell
Paul_Winchell
Discrete analog of a derivative
general, exist. The Newton series, together with the Stirling series and the Selberg series, is a special case of the general difference series, all of which
Finite_difference
German mathematician (1896–1981)
the greatest mathematician of the first half of the 20th century. Atle Selberg said of Siegel and his work: He was in some ways, perhaps, the most impressive
Carl_Ludwig_Siegel
Complex-differentiable part of a Maass wave function
Andrews, Selberg, Hickerson, Choi, McIntosh, and others, who proved Ramanujan's statements about them and found several more examples and identities. (Most
Mock_modular_form
Duality between the process of restricting and inducting in representation theory
overview of the subject of group representations. See Selberg trace formula and the Arthur-Selberg trace formula for generalizations to discrete cofinite
Frobenius_reciprocity
Serbian mathematician and university teacher
Tanigawa, Yoshio (1999). "On Riesz means of the coefficients of the Rankin-Selberg series". Mathematical Proceedings of the Cambridge Philosophical Society
Aleksandar_Ivić
general linear diophantine equation. 628 – Brahmagupta gives Brahmagupta's identity and solves the so-called Pell's equation using his composition method.
Timeline_of_number_theory
Defunct healthcare system, formerly based in the United States
& Policy. March 2, 2010. Retrieved March 25, 2026. Jessee, William F.; Selberg, Jeffrey D. (August 16, 2009). "Exempla explains suit". Daily Camera. Boulder
SCL_Health
Mathematics term
Z) has property (τ) with respect to principal congruence subgroups, by Selberg's theorem. Noncompact solvable groups. Nontrivial free groups and free abelian
Kazhdan's_property_(T)
Supernatural being in Germanic folklore
pp. 172–175. Shippey (2005), pp. 161–68. Alver, Bente Gullveig [no]; Selberg, Torunn (1987), "Folk Medicine as Part of a Larger Concept Complex", Arv
Elf
Type of generalization of periodic functions in Euclidean space
automorphic function is an automorphic form for which j {\displaystyle j} is the identity. An automorphic form is a function F on G (with values in some fixed finite-dimensional
Automorphic_form
Mathematical series
D(u, s) denotes the Dirichlet series of u(n). It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is
Dirichlet_series
cultural landscape, these churches endure as emblematic symbols of enduring identity and heritage, encapsulating the nuanced interplay between religious, mythological
Nordic_folklore
Mathematical functions having established names and notations
essentially complete, such as that of Tannery and Molk, expounded all the basic identities of the theory using techniques from analytic function theory (based on
Special_functions
Generalization of gamma distribution to multiple dimensions
marginalized to yield the density of a single eigenvalue, by evaluating a Selberg integral. The spectral density can be integrated to give the probability
Wishart_distribution
American medical professor
doi:10.1001/jama.2019.1211. ISSN 1538-3598. PMID 31180461. S2CID 182949313. Selberg, Jeff; Prabhjot, Singh; Alday, Jorge (May 16, 2017). "Accelerating Adoption
Prabhjot_Singh_(physician)
communication as it relates to living things and machines. 1948 – Atle Selberg and Paul Erdős prove independently in an elementary way the prime number
Timeline_of_mathematics
2020 edition of IFFR
War Is Never Over, which showcased the selection of work about female identity, power and sexuality, curated by Beth B, Marion Hänsel, à la vie, which
49th International Film Festival Rotterdam
49th_International_Film_Festival_Rotterdam
Prize in Mathematics (first postwar award): Laurent Schwartz and Atle Selberg Nobel Prizes Physics – Cecil Frank Powell Chemistry – Otto Paul Hermann
1950_in_science
Branch of mathematics that studies abstract algebraic structures
2)\quad g_{1}\cdot (g_{2}\cdot v)=(g_{1}g_{2})\cdot v} where e is the identity element of G and g1g2 is the group product in G. The definition for associative
Representation_theory
matrix argument", J. Approx. Theory, 59, no. 2, 224–246, 1989. J. Kaneko, "Selberg Integrals and hypergeometric functions associated with Jack polynomials"
Hypergeometric function of a matrix argument
Hypergeometric_function_of_a_matrix_argument
Academic journal
Cinema. Wiley-Blackwell. p. 321. ISBN 978-1-118-95535-2. OCLC 1269413807. Selberg, Scott (2017). "Rhinestone Cowboy: Alzheimer's, Celebrity, and the Collusions
Celebrity_Studies
Overview of music traditions in Minneapolis, Minnesota, United States
Condition, and Sounds of Blackness have also contributed to the city's musical identity. The city's influence spans multiple genres, including the American folk
Music_of_Minneapolis
transcendental). The conjecture was considered in the early 1940s by Atle Selberg who never formally stated the conjecture. A special case of the conjecture
Four_exponentials_conjecture
Swedish football club
Everyone wanted to work with their own sport leaving football to maintain the identity of Sävast AIF. Sävast AIF has at most run 17 teams catering for approximately
Sävast_AIF
American legal scholar
discrimination-based models toward a more substantive vision of equality. According to Selberg and Wegerstad, Fundamental to Fineman's scholarly work is a feminist critique
Martha_Albertson_Fineman
\operatorname {Ind} _{H}^{G}W} an irreducible representation of G? Maass–Selberg Maass–Selberg relations. matrix coefficient A matrix coefficient of a representation
Glossary of representation theory
Glossary_of_representation_theory
Representation theory
by a cocompact (or cofinite) discrete subgroup. The original paper of Selberg (1956) implicitly invokes the spherical transform; it was Godement (1957)
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
SELBERGS IDENTITY
SELBERGS IDENTITY
SELBERGS IDENTITY
SELBERGS IDENTITY
SELBERGS IDENTITY
SELBERGS IDENTITY
SELBERGS IDENTITY
SELBERGS IDENTITY
SELBERGS IDENTITY