2022/10/24 by Wing Hong Leung, Leung, Wing Hong
Mathematics · #11F30 #11M41 #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2210.13081
openalex publication_date 2022/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A(1,m) be the Fourier coefficients of a SL(3,ℤ) Hecke-Maass cusp form π1 and λ(m) be those of a SL(2,ℤ) Hecke holomorphic or Hecke-Mass cusp form π2. Let H⊂[ [ -X1-ε,X1+ε] ] and \a(h)\h∈ H⊂ℂ be a sequence. We show that if H⊂ ℓ+[ [ 0,X1/2+ε] ] for some ℓ≥0, Da,H(X):=(1)/(|H|)∑h∈ Ha(h)∑m=1^∞ A(1,m)λ(rm+h)V((m)/(X))≪π1,π2,ε \fracX1+ε|H|‖a‖2 for any ε>0, and a similar bound when |H| is big. This improves Sun's bound and generalizes it to an average with arbitrary weights. Moreover, we demonstrate how one can recover the factorizable moduli structure given by the Jutila's circle method via studying a shifted sum with weighted average. This allows us to recover Munshi's bound on the shifted sum with a fixed shift without using the Jutila's circle method.