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We study the mean value estimate of coefficients of product L-function related to a primitive holomorphic cusp form f(z) of weight k for the full modular group SL2(Z). Upper bounds and asymptotic formulas are established. [ABSTRACT FROM AUTHOR]

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]

Assume that α , β are arbitrary arithmetic functions, and that γ is an arithmetic function, which does not correlate with additive characters. By introducing a simple argument, we are able to give a general upper bound for the triple correlation ∑ n ∈ ( X , 2 X ] α ( n ) β ( n − h ) γ ( n + h ) averaging over a short variable h . More precise estimates can be obtained by taking γ to be Fourier coefficients of cusp forms, which improve and streamline the results of Lin and Singh. [ABSTRACT FROM AUTHOR]

We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty [24] to sequences of complex numbers. We further generalize this result when the sequence is comprised of the coefficients of an L -function. As immediate applications, we prove that there are sign changes in intervals within sequences of coefficients of GL ( 2 ) holomorphic cusp forms, GL ( 2 ) Maass forms, and GL ( 3 ) Maass forms. Building on [13,14] , we prove that there are sign changes in intervals within sequences of partial sums of coefficients of GL ( 2 ) holomorphic cusp forms and Maass forms. [ABSTRACT FROM AUTHOR]

Abstract Let π and π′ be irreducible cuspidal automorphic representations of GL (픸) and GL (픸), respectively. Denote by a ( n) the nth coefficient of the Dirichlet series of the Rankin-Selberg L-function associated with π and π′. In this paper, we obtain the upper bound for the sum Σ a ( n). [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 f and g be weight k holomorphic cusp forms and let S f ( n ) and S g ( n ) denote the sums of their first n Fourier coefficients. Hafner and Ivić [9] proved asymptotics for ∑ n ≤ X | S f ( n ) | 2 and proved that the Classical Conjecture, that S f ( X ) ≪ X k − 1 2 + 1 4 + ϵ , holds on average over long intervals. In this paper, we introduce and obtain meromorphic continuations for the Dirichlet series D ( s , S f × S g ) = ∑ S f ( n ) S g ( n ) ‾ × n − ( s + k − 1 ) and D ( s , S f × S g ‾ ) = ∑ n S f ( n ) S g ( n ) n − ( s + k − 1 ) . We then prove asymptotics for the smoothed second moment sums ∑ S f ( n ) S g ( n ) ‾ e − n / X , giving a smoothed generalization of [9] . We also attain asymptotics for analogous sums of normalized Fourier coefficients. Our methodology extends to a wide variety of weights and levels, and comparison with [4] indicates very general cancellation between the Rankin–Selberg L -function L ( s , f × g ) and convolution sums of the coefficients of f and g . In forthcoming works, the authors apply the results of this paper to prove the Classical Conjecture on | S f ( n ) | 2 is true on short intervals, and to prove sign change results on { S f ( n ) } n ∈ N . [ABSTRACT FROM AUTHOR]

In this paper, we establish a general summation formula for the coefficients of a class of L -functions, without assuming the generalized Ramanujan conjecture. As an application, we consider integral power moments of Fourier coefficients of Hecke–Maass cusp forms. [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]

The study of Fourier coefficients of meromorphic modular forms dates back to Ramanujan, who, together with Hardy, studied the reciprocal of the weight 6 Eisenstein series. Ramanujan conjectured a number of further identities for other meromorphic modular forms and quasi-modular forms which were subsequently established by Berndt, Bialek, and Yee. In this paper, we place these identities into the context of a larger family by making use of Poincaré series introduced by Petersson and a new family of Poincaré series which we construct here and which are of independent interest. In addition we establish a number of new explicit identities. In particular, we give the first examples of Fourier expansions for meromorphic modular form with third-order poles and quasi-meromorphic modular forms with second-order poles. [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]

In this paper, we study the exponential sums involving Fourier coefficients of cusp forms for SL ( 2 , Z ) over primes. We are able to establish some new results. In addition, similar results have been obtained for the coefficients of symmetric-square lifts. [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]

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]

For any number m≡0,1(4), we correct the generating function of Hurwitz class number sums ∑rH(4n-mr2) to a modular form (or quasimodular form if m is a square) of weight two for the Weil representation attached to a binary quadratic form of discriminant m and determine its behavior in the Petersson scalar product. This modular form arises through holomorphic projection of the zero-value of a nonholomorphic Jacobi Eisenstein series of index 1 / m. When m is prime, we recover the classical Hirzebruch-Zagier series whose coefficients are intersection numbers of curves on a Hilbert modular surface. Finally, we calculate certain sums over class numbers by comparing coefficients with an Eisenstein series. [ABSTRACT FROM AUTHOR]