# Selberg's 1/4 conjecture

In mathematics, **Selberg's conjecture**, also known as **Selberg's eigenvalue 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 GL_{2} 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_{2}(**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 (ed.),
*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 GL
_{4}and the symmetric fourth of GL_{2}. 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,**VIII**, Providence, R.I.: American Mathematical Society, pp. 1–15, ISBN 978-0-8218-1408-6, MR 0182610