On some estimates involving Fourier coefficients of Maass cusp forms

2022/02/22 by Qing‐Feng Sun, Hui Wang, Sun, Qingfeng +1
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2202.10759

openalex publication_date 2022/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f be a Hecke-Maass cusp form for \rm SL₂(ℤ) with Laplace eigenvalue λ_f(Δ)=1/4+μ² and let λ_f(n) be its n-th normalized Fourier coefficient. It is proved that, uniformly in α, β∈ ℝ, ∑_{n ≤ X}λ_f(n)e(αn²+βn) ≪ X^7/8+ελ_f(Δ)^1/2+ε, where the implied constant depends only on ε. We also consider the summation function of λ_f(n) and under the Ramanujan conjecture we are able to prove ∑_{n ≤ X}λ_f(n)≪ X^1/3+ελ_f(Δ)^4/9+ε with the implied constant depending only on ε.

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