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Let [Formula: see text] be a normalized holomorphic cusp form for [Formula: see text] of weight [Formula: see text] with [Formula: see text]. By the Kuznetsov trace formula for [Formula: see text], we obtain the twisted first moment of the central values of [Formula: see text], where [Formula: see text] varies over Hecke–Maass cusp forms for [Formula: see text]. As an application, we show that such [Formula: see text] is determined by [Formula: see text] as [Formula: see text] varies.

We introduce a new type of summation formula for central values of GL ( 4 ) × GL ( 2 ) L-functions, when varied over Maas forms. By rewriting such a sum in terms of GL ( 4 ) × GL ( 1 ) L-functions and applying a new “balanced” Voronoi formula, the sum can be shown to be equal to a differently-weighted average of the same quantities. By controlling the support of the spectral weighting functions on both sides, this reciprocity formula gives estimates on spectral sums that were previously obtainable only for lower rank groups. The “balanced” Voronoi formula has Kloosterman sums on both sides, and can be thought of as the functional equation of a certain double Dirichlet series involving Kloosterman sums and GL ( 4 ) Hecke eigenvalues. As an application we show that for any self-dual cusp form Π for SL ( 4 , Z ) , there exist infinitely many Maas forms π for SL ( 2 , Z ) such that L ( 1 / 2 , Π × π ) ≠ 0 .

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 λf (n). Let a(n) be the function with squarefull kernel. In this paper, we establish that $$ {\sum}_{n\leqslant x}a(n){\lambda}_f^2\left(n+1\right)= Cx+O\left({x}^{13/14+\upepsilon}\right) $$, where C is a constant that can be explicitly evaluated.

Let f(z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth Hecke eigenvalue or normalized Fourier coefficient by \(\lambda_{f}(n)\). Let g(z) be another distinct primitive holomorphic cusp form of even integral weight \(\ell\) for the full modular group. In this paper, we establish that the set $$\{p| \lambda_f(p^j)

Let F be a self-dual Hecke-Maass cusp form for S L ( 3 , Z ) and let a F ( 1 , n ) denote the n-th coefficient of the Godement-Jacquet L-function L ( s , F ) . Then we show that there exists an absolute constant c 0 > 0 such that ∑ n ⩽ X a F ( 1 , n ) μ ( n ) e ( n α ) ≪ X exp ⁡ ( − c 0 log ⁡ X ) . Here the implied constant depends only on the form F and the bound is uniform in α ∈ R . Moreover, we notice that the aforementioned result generalises to self-dual automorphic cuspidal representations of G L 3 ( A Q ) , with unitary central character.

Let $\mathfrak{q}>2$ be a prime number, $\chi$ a primitive Dirichlet character modulo $\mathfrak{q}$ and $f$ a primitive holomorphic cusp form or a Hecke-Maass cusp form of level $\mathfrak{q}$ and trivial nebentypus. We prove the subconvex bound $$ L(1/2,f\otimes \chi)\ll \mathfrak{q}^{1/2-1/12+\varepsilon}, $$ where the implicit constant depends only on the archimedean parameter of $f$ and $\varepsilon$. The main input is a modifying trivial delta method developed in [1]., Comment: 12 pages

In this article, we have considered the problem of first sign change of Fourier coefficients of a primitive cusp form at a sparse sequence, where elements of the sparse sequence can be written as a sum of two squares. Following the methods of Kohnen–Sengupta, we give an effective upper bound on elements of such a sparse sequence to have first sign change.

Given a fixed Hermitian Jacobi cusp form, we define a family of linear operators between spaces of Hermitian Jacobi cusp forms using Rankin--Cohen brackets. We compute the adjoint maps of such a family with respect to the Petersson scalar product. The Fourier coefficients of the Hermitian Jacobi cusp forms constructed using this method involve special values of certain Dirichlet series associated to Hermitian Jacobi cusp forms.

Let F be the normalized Hecke-eigen cusp form for the full modular group and ζ(s,F) be the corresponding zeta-function. In the paper, the joint universality theorem on the approximation of a collection of analytic functions by shifts (ζ(s+ih1τ,F),…,ζ(s+ihrτ,F)) is proved. Here, h1,…,hr are algebraic numbers linearly independent over the field of rational numbers.

Let $g$ be a Hecke cusp form of half-integral weight, level $4$ and belonging to Kohnen's plus subspace. Let $c(n)$ denote the $n$th Fourier coefficient of $g$, normalized so that $c(n)$ is real for all $n \geq 1$. A theorem of Waldspurger determines the magnitude of $c(n)$ at fundamental discriminants $n$ by establishing that the square of $c(n)$ is proportional to the central value of a certain $L$-function. The signs of the sequence $c(n)$ however remain mysterious. Conditionally on the Generalized Riemann Hypothesis, we show that $c(n) < 0$ and respectively $c(n) > 0$ holds for a positive proportion of fundamental discriminants $n$. Moreover we show that the sequence $\{c(n)\}$ where $n$ ranges over fundamental discriminants changes sign a positive proportion of the time. Unconditionally, it is not known that a positive proportion of these coefficients are non-zero and we prove results about the sign of $c(n)$ which are of the same quality as the best known non-vanishing results. Finally we discuss extensions of our result to general half-integral weight forms $g$ of level $4 N$ with $N$ odd, square-free., 50 pages