Selberg Class

The Selberg class is an axiomatically defined collection of Dirichlet series that captures the essential analytic properties shared by the Riemann zeta function and the most important L-functions of analytic number theory. Introduced by Atle Selberg in 1989, the class is characterized by a small set of requirements — an Euler product, a functional equation, analytic continuation, and a Ramanujan-type bound on coefficients — that together imply a surprising wealth of structure.

The ambition behind the Selberg class is not merely to generalize the zeta function but to identify the minimal conditions from which its deepest properties follow. The Selberg orthogonality conjecture, for example, predicts that distinct members of the class are statistically independent in a precise sense, and the Grand Riemann Hypothesis asserts that all functions in the Selberg class satisfy the analogue of the Riemann hypothesis: their non-trivial zeros all lie on the critical line.

The class connects to the Langlands program through the conjecture that every member of the Selberg class arises from an automorphic representation, and to the theory of functional equations through its central symmetry. Whether the Selberg class is the natural boundary of analytic number theory — or whether there exist important L-functions that escape its axioms — remains an open question that shapes the field's research agenda.