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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 K3 be a non-normal cubic extension over ℚ:We study the higher moment of the coefficients of Dedekind zeta function over sum of two squares where 2 ⩽ l ⩽ 8 and n1; n2; l∈ ℤ. [ABSTRACT FROM AUTHOR]

Let f be a full-level cusp form for GLm(ℤ) with Fourier coefficients Af(cm-2,..., c1, n): Let λ(n) be either the von Mangoldt function Λ(n) or the k-th divisor function τk(n): We consider averages of shifted convolution sums of the type Σ|h|⩽H | ΣX

Let f be a Hecke-Maass cusp form for SL(3,ℤ) with Fourier coeffcients Af (m, n), and let ϕ(x) be a C∞-function supported on [1, 2] with derivatives bounded by ϕ(j)(x)≪j 1: We prove an asymptotic formula for the nonlinear Σn≡lmodqAf (m, n)ϕ(n/X)e(3(kn)1/3/q), where e(z) = e2πiz and k ∈ℤ+. [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]

For a cubic algebraic extension K of ℚ, the behavior of the ideal counting function is considered in this paper. More precisely, let a ( n) be the number of integral ideals of the field K with norm n, we prove an asymptotic formula for the sum $$\sum\nolimits_{n_1^2 + n_2^2 \leqslant x} {a_K \left( {n_1^2 + n_2^2 } \right)} $$ . [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]

Let L( s, sym f) be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL(2,ℤ), with t( n, 1) denoting the nth coefficient of the Dirichlet series for it. It is proved that, for N ⩾ 2 and any α ∈ ℝ, there exists an effective positive constant c such that ΣΛ( n) t( n, 1) e( nα) ≪ N $\exp ( - c\sqrt {\log N} )$, where Λ( n) is the von Mangoldt function, and the implied constant only depends on f. We also study the analogue of Vinogradov's three primes theorem associated to the coefficients of Rankin-Selberg L-functions. [ABSTRACT FROM AUTHOR]

For the normalized Fourier coefficients of Maass cusp forms λ( n) and the normalized Fourier coefficients of holomorphic cusp forms a( n), we give the bound of $\sum\nolimits_{m_1^2 + m_2^2 + m_3^2 \leqslant x} {\lambda (m_1^2 + m_2^2 + m_3^2 )\Lambda (m_1^2 + m_2^2 + m_3^2 )} $ and $\sum\nolimits_{m_1^2 + m_2^2 + m_3^2 \leqslant x} {a(m_1^2 + m_2^2 + m_3^2 )\Lambda (m_1^2 + m_2^2 + m_3^2 )} $. [ABSTRACT FROM AUTHOR]

We study the exponential sums involving Fourier coefficients of Maass forms and exponential functions of the form e( αn), where 0 ≠ α ∈ ℝ and 0 < β < 1. An asymptotic formula is proved for the nonlinear exponential sum Σ λ( n) e( αn), when β = 1/2 and | α| is close to 2√ q, q ∈ ℤ, where λ( n) is the normalized n-th Fourier coefficient of a Maass cusp form for SL(ℤ). The similar natures of the divisor function τ ( n) and the representation function r( n) in the circle problem in nonlinear exponential sums of the above type are also studied. [ABSTRACT FROM AUTHOR]