A curated, non-commercial reference index of freely available mathematics research papers on the Selberg Conjecture — linking directly to arXiv, ResearchGate, and Google Scholar.
Background context and thematic guide to the research areas covered in this index.
The Selberg Conjecture (also known as Selberg's Eigenvalue Conjecture) was proposed by Atle Selberg in 1965. It asserts that for any congruence subgroup Γ of SL(2,ℤ), the smallest non-zero eigenvalue λ₁ of the Laplace–Beltrami operator on the associated hyperbolic surface satisfies λ₁ ≥ 1/4.
This conjecture is deeply connected to the Ramanujan Conjecture for GL(2) automorphic forms, and more broadly to the Generalized Ramanujan Conjecture (GRC) for higher-rank groups. It sits at the intersection of spectral theory, automorphic forms, and analytic number theory.
The best known result toward the conjecture is due to Kim and Sarnak (2003), who established λ₁ ≥ 975/4096 — derived from Kim's proof of the functoriality of the symmetric fourth power lift of automorphic forms on GL(2).
Eigenvalues of the Laplacian on hyperbolic surfaces and congruence subgroups.
Maass forms, cusp forms, and their Fourier coefficients on GL(n).
Connections between Selberg's eigenvalue bound and the Ramanujan–Petersson conjecture.
Langlands functoriality, symmetric power lifts, and their role in bounding eigenvalues.
Sieve methods, L-functions, and applications of spectral gaps to prime distribution.
Selberg's original trace formula, zeta functions, and early progress on the conjecture.
References sourced from the Wikipedia article on the Selberg Conjecture. No papers are hosted here. Search arXiv, ResearchGate, or Google Scholar directly using the titles and authors below.
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