Selberg's 1/4 conjecture

In mathematics, Selberg's conjecture, conjectured by Selberg (1965, p. 13), 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.

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 GL₂ over the rationals at the infinite place, and says that the component at infinity of the corresponding representation is a principal series representation of GL₂(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.

References

Gelbart, S. (2001) [1994], "Selberg's 1/4 conjecture", Encyclopedia of Mathematics, EMS Press
Kim, Henry H.; Sarnak, Peter (2003), "Functoriality for the exterior square of GL₄ and the symmetric fourth of GL₂. Appendix 2.", Journal of the American Mathematical Society, 16 (1): 139–183, doi:10.1090/S0894-0347-02-00410-1, ISSN 0894-0347, MR 1937203
Selberg, Atle (1965), "On the estimation of Fourier coefficients of modular forms", in Whiteman, Albert Leon (ed.), Theory of Numbers, Proceedings of Symposia in Pure Mathematics, vol. VIII, Providence, R.I.: American Mathematical Society, pp. 1–15, ISBN 978-0-8218-1408-6, MR 0182610