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], "s/s130210", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
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, ISSN 0894-0347, MR 1937203, doi:10.1090/S0894-0347-02-00410-1
Selberg, Atle (1965), "On the estimation of Fourier coefficients of modular forms", in Whiteman, Albert Leon, Theory of Numbers, Proceedings of Symposia in Pure Mathematics, VIII, Providence, R.I.: American Mathematical Society, pp. 1–15, ISBN 978-0-8218-1408-6, MR 0182610

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