Sums of Fourier coefficients of cusp forms of level <Emphasis Type="Italic">D</Emphasis> twisted by exponential functions |
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Authors: | Huan Liu |
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Institution: | 1.School of Mathematics,Shandong University,Jinan,China |
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Abstract: | Let g be a holomorphic or Maass Hecke newform of level D and nebentypus χD, and let a g (n) be its n-th Fourier coefficient. We consider the sum \({S_1} = \sum {_{X < n \leqslant 2X}{a_g}\left( n \right)e\left( {\alpha {n^\beta }} \right)}\) and prove that S 1 has an asymptotic formula when β = 1/2 and α is close to \(\pm 2\sqrt {q/D}\) for positive integer q ≤ X/4 and X sufficiently large. And when 0 < β < 1 and α, β fail to meet the above condition, we obtain upper bounds of S 1. We also consider the sum \({S_2} = \sum {_{n > 0}{a_g}\left( n \right)e\left( {\alpha {n^\beta }} \right)\phi \left( {n/X} \right)}\) with ø(x) ∈ C c ∞ (0,+∞) and prove that S 2 has better upper bounds than S 1 at some special α and β. |
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