Sums of Fourier coefficients involving theta series and Dirichlet characters

2024/10/16 by Yanxue Yu, Yu, Yanxue
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2410.12305

openalex publication_date 2024/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f be a holomorphic or Maass cusp forms for \rm SL₂(ℤ) with normalized Fourier coefficients λ_f(n) and \bna r_ℓ(n)=#\(n₁,⋯,n_ℓ)∈ ℤ²:n₁²+⋯+n_ℓ²=n\. \ena Let χ be a primitive Dirichlet character of modulus p, a prime. In this paper, we are concerned with obtaining nontrivial estimates for the sum \bna ∑_n≥1λ_f(n)r_ℓ(n)χ(n)w((n)/(X)) \ena for any ℓ ≥ 3, where w(x) be a smooth function compactly supported in [1/2,1].

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