What is the Selberg class of L-functions?

The Selberg class is an axiomatic definition capturing the essential properties shared by most functions known as L-functions or zeta functions. Introduced by Atle Selberg, its members are Dirichlet series obeying four specific axioms. Researchers hope that defining this class will lead to a full classification of its contents and deeper insight into the Riemann hypothesis.

What we know

• The Selberg class is an axiomatic definition of a class of L-functions composed of Dirichlet series obeying four axioms. [1, 2]
• The class was defined by Atle Selberg in 1992, though he preferred not to use the word axiom. [2]
• A primary hope for the Selberg class definition is that it will lead to a classification of its contents and insights into the Riemann hypothesis. [2]

Still open

• The exact nature of the Selberg class remains conjectural.

Where to go next

• the bigger pictureWhat is an L-function?
• how it workedHow did Atle Selberg define the trace formula?
• a case in pointWhat is the Selberg zeta function?
• what followedHow does the Riemann hypothesis relate to zeta functions?
• whoWho was Atle Selberg?
• an unexpected connectionHow do lengths of closed geodesics relate to prime numbers?The Selberg zeta function uses lengths of simple closed geodesics instead of traditional prime numbers to form its product expansion.

Sources

• [1]Selberg class (Wikidata Q7447524) · CC0 1.0
• [2]Selberg class · CC BY-SA 4.0
• [3]L-function · CC BY-SA 4.0
• [4]Selberg zeta function · CC BY-SA 4.0
• [5]Selberg trace formula · CC BY-SA 4.0