2023/11/23 by Dimitrios Chatzakos, Chatzakos, Dimitrios, Corentin Darreye +3
Mathematics · #11F12 #11F72 (primary) #58J51 #81Q50 (secondary) #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2311.14184
openalex publication_date 2023/11/23 · openalex created_date 2023/11/28 · openalex updated_date 2026/07/28
We investigate the mean value of the inner product of squared GLn degenerate maximal parabolic Eisenstein series against a smooth compactly supported function lying in a restricted space of incomplete Eisenstein series induced from a SL2(ℤ) Hecke-Maass cusp form φ. Our result breaks the fundamental threshold with a polynomial power-saving beyond the pointwise implications of the generalised Lindelöf hypothesis for L-functions attached to φ. Furthermore, we evaluate the archimedean quantum variance and establish approximate orthogonality, expanding upon Zhang's (2019) work on quantum unique ergodicity for GLn degenerate maximal parabolic Eisenstein series as well as Huang's (2021) work on quantum variance for GL2 Eisenstein series. Despite the theoretical strength of these manifestations, our argument relies exclusively on the Watson-Ichino-type formula for incomplete Eisenstein series of type (2, 1, …, 1) and Jutila's (1996) asymptotic formula for the second moment of L-functions attached to φ in long intervals, supplemented by a standard analytical toolbox.