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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 nonzero cusp form of even integral weight k ⩾ 2 on the Hecke congruence subgroup Γ 0 ( N ) with N squarefree. Suppose that the normalized Fourier coefficients λ f ( n ) of f are real. We prove that the first sign change of λ f ( n ) occurs in the range n ≪ ( k N ) 2 + ε . This improves upon the earlier result of Choie and Kohnen [2] . [ABSTRACT FROM AUTHOR]

With the method of moments and the mollification method, we study the central L -values of GL(2) Maass forms of weight 0 and level 1 and establish a positive-proportional nonvanishing result of such values in the aspect of large spectral parameter in short intervals, which is qualitatively optimal in view of Weyl's law. As an application of this result and a formula of Katok–Sarnak, we give a nonvanishing result on the first Fourier coefficients of Maass forms of weight 1 2 and level 4 in the Kohnen plus space. [ABSTRACT FROM AUTHOR]

We construct certain Casimir operators and study the spectral properties of their resolvents on L 2 ( Γ \ Sp 2 ( R ) ) . We define non-holomorphic multi-variable Poincaré series of exponential type for symplectic groups and continue them analytically in case of genus two for the small weight four using the above resolvents. We apply our results to describe the holomorphic projection to the weight four holomorphic discrete series in terms of Fourier coefficients by using Sturm's operator. This paper is a study of a prototype for symplectic groups and special orthogonal groups. [ABSTRACT FROM AUTHOR]

Let f be a weight k holomorphic cusp form of level one, and let S f ( n ) denote the sum of the first n Fourier coefficients of f . In analogy with Dirichlet's divisor problem, it is conjectured that S f ( X ) ≪ X k − 1 2 + 1 4 + ϵ . Understanding and bounding S f ( X ) has been a very active area of research. The current best bound for individual S f ( X ) is S f ( X ) ≪ X k − 1 2 + 1 3 ( log ⁡ X ) − 0.1185 from Wu [17] . Chandrasekharan and Narasimhan [2] showed that the Classical Conjecture for S f ( X ) holds on average over intervals of length X . Jutila [10] improved this result to show that the Classical Conjecture for S f ( X ) holds on average over short intervals of length X 3 4 + ϵ . Building on the results and analytic information about ∑ | S f ( n ) | 2 n − ( s + k − 1 ) from our recent work [9] , we further improve these results to show that the Classical Conjecture for S f ( X ) holds on average over short intervals of length X 2 3 ( log ⁡ X ) 1 6 . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study products of Fourier coefficients of an elliptic cusp form, () for a fixed positive integer r, concerning both non-vanishing and non-negativity. [ABSTRACT FROM AUTHOR]

Let f be a Hecke eigencusp form of even integral weight k or Maass cusp form for the full modular group S L 2 ( Z ) . Denote by λ sym m f ( n ) the n th normalized coefficient of the Dirichlet expansion of the m th symmetric power L -function associated to f . In this paper, we establish some bounds for ∑ n ⩽ x λ sym m f ( n ) , ∑ n ⩽ x λ f ( n m ) , which improve the corresponding results of Lau & Lü [8] . [ABSTRACT FROM AUTHOR]

Let f be a holomorphic Hecke eigenform of weight k for the modular group Γ = SL (Z) and let λ ( n) be the n-th normalized Fourier coefficient. In this paper, by a new estimate of the second integral moment of the symmetric square L-function related to f, the estimate is established, which improves the previous result. [ABSTRACT FROM AUTHOR]

Let g ( z ) be a Maass cusp form for SL ( 2 , Z ) , and let λ g ( n ) be the n -th Fourier coefficient of g ( z ) . In this paper we investigate the nonlinear exponential sum ∑ n ∼ X n ≡ l mod q λ g ( n ) e ( α n β ) twisted by Fourier coefficient over the arithmetic progression. We prove an asymptotic formula when β = 1 / 2 and α is close to ± 2 k q , k ∈ Z + . [ABSTRACT FROM AUTHOR]

In this note, we first see that the isomorphism in [26] between vector-valued modular forms and scalar-valued modular forms holds for more general discriminant forms. With this established, we shall prove the Zagier duality for canonical bases for spaces of weakly holomorphic modular forms satisfying some ϵ -condition, thus obtain the complete grids for the duality. Finally we raise a question on certain integrality of the Fourier coefficients of reduced modular forms and provide a partial solution. [ABSTRACT FROM AUTHOR]

Let f( z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth normalized Fourier coefficient (Hecke eigenvalue) by λ( n). Let d( n) be the Dirichlet divisor function. In this paper, we establish that where $${0 < \delta_{j} < 1}$$ , $${0 < \delta_{i,j}\leqq 1}$$ ( $${\delta_{i,j} = 1}$$ if and only if i = j), and . [ABSTRACT FROM AUTHOR]

Let F be a totally real field and χ a primitive narrow ray class character of F . We prove a formula for the Fourier coefficients of the Siegel Eisenstein series of degree n , weight k and character χ under certain conditions on χ . We also prove the existence of Λ -adic Siegel Eisenstein series of degree n . [ABSTRACT FROM AUTHOR]

We formulate an analogue of the Ichino–Ikeda conjectures for the Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups. This sharpens the conjectures of Sakellaridis–Venkatesh in the case at hand. [ABSTRACT FROM AUTHOR]

We characterize all cusp forms among the degree two Siegel modular forms by the growth of their Fourier coefficients. We also give a similar result for Jacobi forms over the group . [ABSTRACT FROM AUTHOR]

Let λf( n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f( z) ∈ Sk(Γ). In this paper, we established nontrivial estimates for $$\sum\limits_{n \leqslant x} {\lambda _f \left( {n^i } \right)\lambda _f \left( {n^j } \right),}$$ where 1 ≤ i < j ≤ 4. [ABSTRACT FROM AUTHOR]