The highest lowest zero and other applications of positivity

DOI10.1215/S0012-9074-02-11213-7zbMath1014.11036arXivmath/0112196OpenAlexW2050879214MaRDI QIDQ1847907

Stephen D. Miller

Publication date: 27 October 2002

Published in: Duke Mathematical Journal (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0112196

zbMATH Keywords

Riemann zeta functioncohomology groupsEisenstein series\(L\)-functionsgeneralized Riemann hypothesisautomorphic formslower boundsvanishing theoremseigenvalues of the Laplaciantest functionnoncuspidal automorphic formRamanujan-Selberg conjectureRiemann-Weil explicit formula

Mathematics Subject Classification ID

Other Dirichlet series and zeta functions (11M41) Langlands (L)-functions; one variable Dirichlet series and functional equations (11F66) Nonreal zeros of (zeta (s)) and (L(s, chi)); Riemann and other hypotheses (11M26) Spectral theory; trace formulas (e.g., that of Selberg) (11F72)

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Cites Work

• Vanishing of the cuspidal cohomology of congruence subgroups of \(\text{GL}_ n(\mathbb{Z})\)
• On Selberg's eigenvalue conjecture
• Zeros of principal \(L\)-functions and random matrix theory
• Le spectre résiduel de ${\rm GL}(n)$
• Eigenvalues of the Laplacian for Hecke triangle groups
• Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions : a survey of recent results
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