On Selberg's eigenvalue conjecture

DOI10.1007/BF01895672zbMath0844.11038WikidataQ123354425 ScholiaQ123354425MaRDI QIDQ1902003

Wenzhi Luo, Ze'ev Rudnick, Peter C. Sarnak

Publication date: 1 September 1996

Published in: Geometric and Functional Analysis. GAFA (Search for Journal in Brave)

Full work available at URL: https://eudml.org/doc/58194

zbMATH Keywords

error term approximate functional equation Ramanujan conjecture prime geodesic theorem sums of Kloosterman sums Selberg's eigenvalue conjecture Gelbart-Jacquet lift bounds on hyper-Kloosterman sums eigenvalues of the non-Euclidean Laplacian for congruence subgroups Rankin-Selberg theory of $L$-functions twisted Rankin-Selberg $L$-function

Mathematics Subject Classification ID

Estimates of eigenvalues in context of PDEs (35P15) Gauss and Kloosterman sums; generalizations (11L05) Spectral theory; trace formulas (e.g., that of Selberg) (11F72)

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Cites Work