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.

For the conjecture about the Riemann zeta function, see Selberg's zeta function conjecture.

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 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, VIII, Providence, R.I.: American Mathematical Society, pp. 1–15, ISBN 978-0-8218-1408-6, MR 0182610
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