In mathematics, the Selberg class is an axiomatic definition of a class of L-functions. The members of the class are Dirichlet series which obey four axioms...

Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms...

showed that a suitable analog of the Selberg trace formula for the action of the idèle class group on the adèle class space would imply the Riemann hypothesis...

In mathematics, the Selberg trace formula, introduced by Selberg (1956), is an expression for the character of the unitary representation of a Lie group...

In mathematics, the Chowla–Selberg formula is the evaluation of a certain product of values of the gamma function at rational values in terms of values...

In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over...

series, as are the Dirichlet L-functions. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named...

conjecture Riemann hypothesis and the generalized Riemann hypothesis. Selberg class S Explicit formulae for L-functions Trace formula A directory of all...

Q, and Δ its discriminant. Artin's conjecture Dirichlet L-function Selberg class Grand Riemann hypothesis Davenport, Harold (2000). Multiplicative Number...

term L-function here includes many known types of zeta functions. The Selberg class is an attempt to capture the core properties of L-functions in a set...

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...

(Martian crater) List of things named after Niels Henrik Abel Abel Prize Selberg class Weather forecasting Aanderaa–Karp–Rosenberg conjecture "Niels Henrik...

Euler system p-adic L-function Arithmetic geometry Complex multiplication Abelian variety of CM-type Chowla–Selberg formula Hasse–Weil zeta function...

all arise as automorphic L-functions, and hence should be part of the Selberg class. There are also conjectures concerning the values of these L-functions...

of all residue classes of numbers are removed, as opposed to small sieves such as the Selberg sieve wherein only a few residue classes are removed. The...

of L-functions, and in particular are expected to coincide with the Selberg class. Furthermore, all L-functions over arbitrary number fields are widely...

associated with quadratic extensions of the rationals by class field theory. In this case Selberg's trace formula shows that the spectrum of the Laplacian...

conversation with Landau, there was little basis for such speculation. However Selberg in the early 1950s proved a duality between the length spectrum of a Riemann...

(443,449), (457,463), (461,467). D.H.J. Polymath (2014). "Variants of the Selberg sieve, and bounded intervals containing many primes". Research in the Mathematical...

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...

is known for the Arthur–Selberg trace formula, generalizing the Selberg trace formula from the rank-one case (due to Selberg himself) to general reductive...

theorem represents the beginning of rigorous analytic number theory. Atle Selberg (1949) gave an elementary proof. Dirichlet's theorem is proved by showing...

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...

the existence of cusp forms led to the disproof of a conjecture of Atle Selberg. He has obtained the strongest known bounds towards the Ramanujan–Petersson...

finite. In particular, if it is finitely generated then it is finite. Selberg's lemma: any finitely generated linear group contains a torsion-free subgroup...

Selberg. However, the circumstances leading up to the proofs, as well as publication disagreements, led to a bitter dispute between Erdős and Selberg...

overview of the subject of group representations. See Selberg trace formula and the Arthur-Selberg trace formula for generalizations to discrete cofinite...

to say, as has been asserted from time to time in the literature, that Selberg's sieve is always better than Brun's." Halberstam, Heini; Richert, Hans-Egon...

Selberg, A. "On Epstein's Zeta-Function". EuDML. pp. 86–110. Berndt, Bruce C.; Chan, Heng Huat; Zhang, Liang-Cheng (6 June 1997). "Ramanujan's class invariants...

equal to x; then π(x) ~ x/ln(x). This result is usually credited to Atle Selberg. In his 1970 Ph.D. thesis, Grigory Margulis proved a similar result for...