Sums of Cusp Form Coefficients Along Quadratic Sequences

2023/01/27 by Chan Ieong Kuan, David Lowry-Duda, Kuan, Chan Ieong +5
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2301.11901

openalex publication_date 2023/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f(z) = ∑ A(n) n^(k-1)/2 e(nz) be a cusp form of weight k ≥ 3 on Γ₀(N) with character χ. By studying a certain shifted convolution sum, we prove that ∑_{n ≤ X} A(n²+h) = c_f,h X + O_f,h,ε(X^(3)/(4)+ε) for ε>0, which improves a result of Blomer from 2008 with error X^(6)/(7)+ε. This includes an appendix due to Raphael S. Steiner, proving stronger bounds for certain spectral averages.

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