摘要
Let g be a holomorphic or Maass Hecke newform of level D and nebentypus XD, and let ag(n) be its n-th Fourier coefficient. We consider the sum S1=∑ X〈n≤2x ag(n)e(an β) and prove that S1 has an asymptotic formula when β = 1/2 and α is close to ±2 √q/D for positive integer q ≤ X/4and X sufficiently large. And when 0 〈β 〈 1 and α, β fail to meet the above condition, we obtain upper bounds of S1. We also consider the sum S2 = ∑n〉0 ag(n)e(an β) Ф(n/X) with Ф(x) ∈ C c ∞(0,+∞) and prove that S2 has better upper bounds than S1 at some special α and β.
Let g be a holomorphic or Maass Hecke newform of level D and nebentypus XD, and let ag(n) be its n-th Fourier coefficient. We consider the sum S1=∑ X〈n≤2x ag(n)e(an β) and prove that S1 has an asymptotic formula when β = 1/2 and α is close to ±2 √q/D for positive integer q ≤ X/4and X sufficiently large. And when 0 〈β 〈 1 and α, β fail to meet the above condition, we obtain upper bounds of S1. We also consider the sum S2 = ∑n〉0 ag(n)e(an β) Ф(n/X) with Ф(x) ∈ C c ∞(0,+∞) and prove that S2 has better upper bounds than S1 at some special α and β.
基金
This work was supported in part by the Natural Science Foundation of Shandong Province (No. ZR2015AM016).