What is the Selberg class?

The Selberg class is an axiomatic definition of a class of L-functions in mathematics, introduced by Norwegian mathematician Atle Selberg in 1992. Its members consist of Dirichlet series that obey four specific axioms capturing essential properties of common L-functions and zeta functions. Although the exact nature and full classification of the class remain conjectural, it aims to shed light on properties like the Riemann hypothesis.

What we know

• The Selberg class is an axiomatic definition of a class of L-functions composed of Dirichlet series. [3]
• Members of the Selberg class obey four axioms capturing the essential properties of functions commonly called L-functions or zeta functions. [3]
• The class was defined by the Norwegian mathematician Atle Selberg in 1992. [2, 3]
• Researchers hope the definition of the Selberg class will lead to a classification of its contents and give insight into the Riemann hypothesis. [3]

Still open

• The exact nature of the Selberg class is conjectural.

Where to go next

• whoWho was Atle Selberg?
• the bigger pictureWhat is the Riemann hypothesis?
• how it workedWhat is the Arthur-Selberg trace formula?
• compared withHow does analytic number theory relate to spectral theory?
• a case in pointWhat are Dirichlet series?
• an unexpected connectionHow do automorphic forms connect to zeta functions?The Selberg class attempts to capture properties of L-functions and zeta functions to build insight into their relationship with automorphic forms.

Sources

• [1]Atle Selberg (Wikidata Q211041) · CC0 1.0
• [2]Atle Selberg · CC BY-SA 4.0
• [3]Selberg class · CC BY-SA 4.0
• [4]Riemann hypothesis · CC BY-SA 4.0
• [5]Arthur–Selberg trace formula · CC BY-SA 4.0