Beyond the Weyl barrier for GL(2) exponential sums

2021/04/12 by Roman Holowinsky, Holowinsky, Roman, Ritabrata Munshi +3
Mathematics · #11F66(Secondary) #11L07(Primary) 11F30 #Advanced Algebra and Geometry #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2104.05157

openalex publication_date 2021/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we use the Bessel δ-method, along with new variants of the van der Corput method in two dimensions, to prove non-trivial bounds for GL(2) exponential sums beyond the Weyl barrier. More explicitly, for sums of GL(2) Fourier coefficients twisted by e(f(n)), with length N and phase f(n)=N^β log n / 2π or a n^β, non-trivial bounds are established for β< 1.63651... , which is beyond the Weyl barrier at β= 3/2.

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