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 subgroup
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the ...
s 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 975/4096≈0.238..., due to Kim and Sarnak (2003).
The
generalized Ramanujan conjecture for the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
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
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External links
* {{Cite web , title=Selberg conjecture - Encyclopedia of Mathematics , url=https://encyclopediaofmath.org/wiki/Selberg_conjecture , access-date=2022-06-08 , website=encyclopediaofmath.org
Automorphic forms
Conjectures