2001/12/19 by Stephen D. Miller, Miller, Stephen D. · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT) #math.NT #math.RT
paper · pdf · doi:10.48550/arxiv.math/0112196
25 pages, 15 figures. To appear in the Duke Math. Journal. http://www.math.rutgers.edu/~sdmiller
arxiv created 2001/12/19 · openalex publication_date 2001/12/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The first nontrivial zeroes of the Riemann ζ function are ≈ 1/2+±14.13472i. We investigate the question of whether or not any other L-function has a higher lowest zero. To do so we try to quantify the notion that the L-function of a ``small'' automorphic representation (i.e. one with small level and archimedean type) does not have small zeroes, and vice-versa. We prove that many types of automorphic L-functions have a lower first zero than ζ's. This is done using Weil's explicit formula with carefully-chosen test functions. When this method does not immediately show L-functions of a certain type have low zeroes, we then attempt to turn the tables and show no L-functions of that type exist. Thus the argument is a combination of proving low zeroes exist and that certain cusp forms do not. Consequently we are able to prove cohomological vanishing theorems and improve upon existing bounds on the Laplace spectrum on L2(\quo n).