mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Selberg's conjecture, also known as Selberg's eigenvalue conjecture, conjectured by , states that the

eigenvalues In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

of the

Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \ ...

Maass wave form In mathematics, Maass forms or Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup \ ...

s 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 is the subgroup of invertible integer matrices of determinant 1 in which the off-diag ...

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 . The

generalized Ramanujan conjecture A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...

for the

general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...

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 In mathematics, the Langlands program is a set of conjectures about connections between number theory, the theory of automorphic forms, and geometry. It was proposed by . It seeks to relate the structure of Galois groups in algebraic number theo ...

, 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