In mathematics, Selberg's conjecture, also known as Selberg's eigenvalue conjecture, conjectured by , states that the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least 1/4. Selberg showed that the eigenvalues are at least 3/16. Subsequent works improved the bound, and the best bound currently known is 975/4096âÂÂ0.238..., due to .
The generalized Ramanujan conjecture for the general linear group implies Selberg's conjecture. More precisely, Selberg's conjecture is essentially the generalized Ramanujan conjecture for the group GL<sub>2</sub> over the rationals at the infinite place, and says that the component at infinity of the corresponding representation is a principal series representation of GL<sub>2</sub>(R) (rather than a complementary series representation). The generalized Ramanujan conjecture in turn follows from the Langlands functoriality conjecture, and this has led to some progress on Selberg's conjecture.