Related papers: Shifted convolution sums for $GL(3)\times GL(2)$

First we prove a modified version of the famous Lemma on the mean square estimate for exponential sums, by plugging the Cesaro weights in the right hand side of Gallagher's inequality. Then we apply it, in order to establish a mean value…

Let $g$ be a primitive holomorphic or Maass newform for $\Gamma_0(D)$. In this paper, by studying the Bessel integrals associated to $g$, we prove an asymptotic Bessel $\delta$-identity associated to $g$. Among other applications, we prove…

In this paper, we prove a Voronoi summation formula for the shifted 3-fold divisor function twisted by additive characters. As the main tool, we provide the functional equation for the shifted $GL(3)$ Estermann function.

Motivated by problems in control theory concerning decay rates for the damped wave equation $$w_{tt}(x,t) + \gamma(x) w_t(x,t) + (-\Delta + 1)^{s/2} w(x,t) = 0,$$ we consider an analogue of the classical Paneah-Logvinenko-Sereda theorem for…

We investigate the evolution of dark matter halo shapes in cosmological N-body simulations both in scale free Einstein-De Sitter (EdS) and $\Lambda$CDM cosmologies. We compute the axis ratios ($q=b/a,s=c/a$) of well resolved central halos…

Suppose $\pi_1$ and $\pi_2$ are two Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ such that for all primitive character $\chi$ we have $$ L(\tfrac{1}{2},\pi_1\otimes\chi)=L(\tfrac{1}{2},\pi_2\otimes\chi). $$ Then we show that $\pi_1=\pi_2$.

In this paper we study restricted sum formulas involving alternating Euler sums which are defined by \zeta(s_1,...,s_{d};\epsilon_1,...,\epsilon_d)=\sum_{n_1>...>n_d\ge 1}\frac{\epsilon_1^{n_1}... \epsilon_{d}^{n_d}}{n_1^{s_1}...…

In this paper, we prove the following two results. Let $d$ be a natural number and $q,s$ be co-prime integers such that $1 < qs$. Then there exists a constant $\delta > 0$ depending only on $q,s$ and $d$ such that for any finite subset $A$…

Let $\pi$ be a cuspidal representation on $\GL(2,\mathbb{A}_{\mathbb{Q}}).$ We give nontrivial lower and upper bounds for average of absolute values of Dirichlet coefficients associated to $\pi;$ and nontrivial upper bound in the case of…

We derive spectral sum rules for inverse powers of the eigenvalues of the Helmholtz equation on a $d$-sphere in the presence of an arbitrary density. By adopting a rigorous renormalization scheme, we remove the divergent contributions of…

We establish effective versions of Oppenheim's conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result which…

Let $v:[0,T]\times \R^d \to \R$ be the solution of the parabolic backward equation $ \partial_t v + (1/2) \sum_{i,l} [\sigma \sigma^\perp]_{il} \partial_{x_i \partial_{x_l} v + \sum_{i} b_i \partial_{x_i}v + kv =0$ with terminal condition…

We present a noncommutative D=3, N=1 supergravity, invariant under diffeomorphisms, local U(1,1) noncommutative \star-gauge transformations and local \star-supersymmetry. Its commutative limit is the usual D=3 pure supergravity, without…

Consider the trilinear form for twisted convolution on $\mathbb{R}^{2d}$: \begin{equation*} \mathcal{T}_t(\mathbf{f}):=\iint f_1(x)f_2(y)f_3(x+y)e^{it\sigma(x,y)}dxdy,\end{equation*} where $\sigma$ is a symplectic form and $t$ is a…

We develop new closed form representations of sums of (n + {\alpha})th shifted harmonic numbers and reciprocal binomial coefficients in terms of {\alpha}th shifted harmonic numbers. Some interesting new consequences and illustrative…

We use a relative trace formula on GL(2) to compute a sum of twisted modular L-functions anywhere in the critical strip, weighted by a Fourier coefficient and a Hecke eigenvalue. When the weight k or level N is sufficiently large, the sum…

Let $G$ be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to 1, e.g. $G= \SL_2 (\C) \times \SL_2 (\C)$ or $\SL_3 (\C)$. Then the fundamental rank of $G$ is $2,$ and according to the conjecture…

We extend the results obtained by E. Ntienjem to all positive integers. Let $\EuFrak{N}$ be the subset of $\mathbb{N}$ consisting of $\,2^{\nu}\mho$, where $\nu$ is in $\{0,1,2,3\}$ and $\mho$ is a squarefree finite product of distinct odd…

For $ 1\le k <n$, we prove that for functions $F,G$ on $ {\Bbb R}^{n}$, any $k$-dimensional affine subspace $H \subset {\Bbb R}^{n}$, and $p,q,r \ge 2$ with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1$, one has the estimate $$…

In this short note, we treat an unbalanced shifted convolution sum of Fourier coefficients of cusp forms by a rather simple argument. Our result improves previous results established by more advanced approaches.