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Let f be a holomorphic Hecke cusp form with even integral weight k ⩾ 2 for the full modular group, and let χ be a primitive Dirichlet character modulo q. Let Lf (s, χ) be the automorphic L-function attached to f and χ. We study the mean-square estimate of Lf (s, χ) and establish an asymptotic formula. [ABSTRACT FROM AUTHOR]

Let f be a Maass cusp form for Γ( N) with Fourier coefficients λ( n) and Laplace eigenvalue $$\frac{1} {4} + k^2 $$ . For real α ≠ 0 and β > 0, consider the sum S ( f; α, β) = ∑ λ( n) e( αn ) ϕ( n/ X), where ϕ is a smooth function of compact support. We prove bounds for the second spectral moment of S ( f; α, β), with the eigenvalue tending towards infinity. When the eigenvalue is sufficiently large, we obtain an average bound for this sum in terms of X. This implies that if f has its eigenvalue beyond $$X^{\tfrac{1} {2} + \varepsilon } $$ , the standard resonance main term for S ( f; $$ \pm 2\sqrt q $$ , 1/2), q ∈ ℤ, cannot appear in general. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL(2) × GL(2). It contains in particular a proof of an asymptotic expansion of a well-known oscillatory integral with an enlarged range of K ⩽ L ⩽ K . The same bounds can be proved in a similar way for holomorphic cusp forms. [ABSTRACT FROM AUTHOR]

Let g be a holomorphic or Maass Hecke newform of level D and nebentypus χD, and let a ( 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 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 . 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 (0,+∞) and prove that S has better upper bounds than S at some special α and β. [ABSTRACT FROM AUTHOR]