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

Let $\lambda_g (n)$ be the Fourier coefficients of a holomorphic cusp modular form $g$ for $\mathrm{SL}_2 (\mathbb{Z})$. The aim of this article is to get non-trivial bound on non-linearly additively twisted sums of the Fourier coefficients…

Infinite series sum_{n=1}^infty {(alpha/2)_n / (n n!)}_1F_1(-n, gamma, x^2), where_1F_1(-n, gamma, x^2)={n!_(gamma)_n}L_n^(gamma-1)(x^2), appear in the first-order perturbation correction for the wavefunction of the generalized spiked…

Let $\pi$ be a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form, $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or Maass cusp form and $\chi$ be any non-trivial character $\bmod \, p$, where $p$ is prime. We show that the $L$-function associated…

Let $\pi$ be a Hecke-Maass cusp form for $SL(3,\mathbb Z)$ and $f$ be a holomorphic (or Maass) Hecke form for $SL(2,\mathbb{Z})$. In this paper we prove the following subconvex bound $$ L\left(\tfrac{1}{2}+it,\pi\times…

Let $\pi$ be a $SL(3,\mathbb Z)$ Hecke-Maass cusp form, and let $\chi$ be a primitive Dirichlet character modulo $M$, which we assume to be prime. In this note we revisit the subconvexity problem addressed in `The circle method and bounds…

In this work we provide a meromorphic continuation in three complex variables of two types of triple shifted convolution sums of Fourier coefficients of holomorphic cusp forms. The foundations of this construction are based in the…

We establish an estimate on sums of shifted products of Fourier coefficients coming from holomorphic or Maass cusp forms of arbitrary level and nebentypus. These sums are analogous to the binary additive divisor sum which has been studied…

Let $d$ be a probability distribution. Under certain mild conditions we show that $$ \lim_{x\to\infty}x\sum_{n=1}^\infty \frac{d^{*n}(x)}{n}=1,\qquad\text{where}\quad d^{*n}:=\underbrace{\,d*d*\cdots*d\,}_{n\text{ times}}. $$ For a…

In this paper, we study non-trivial upper bounds for the sum $\sum \limits_{n \in S} |\lambda_f(n)|$ where $f$ is a normalized Maass eigencusp form for the full modular group, $\lambda_f(n)$ is the $n$th normalized Fourier coefficient of…

For a nonprincipal character $\chi$ modulo $D$, when $x\ge D^{\frac56+\varepsilon}$, $(l,D) = 1$, we prove a nontrivial estimate of the form $\sum_{n\le x}\Lambda (n)\chi (n-l)\ll x\exp\left(-0.6\sqrt{\ln D}\right)$ for the sum of values of…

We develop certain aspects of the theory of shifted multiple Dirichlet series and study their meromorphic continuations. These continuations are used to obtain explicit spectral first and second moments of Rankin-Selberg convolutions. One…

Let $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or the Eisenstien series $E(z,1/2)$ and $\pi$ be a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form with its Langlands parameter $\mu$ in generic position i.e. away from Weyl chamber walls and…

Let $\pi$ be a $SL(3,\mathbb Z)$ Hecke-Maass cusp form satisfying the Ramanujan conjecture and the Selberg-Ramanujan conjecture, and let $\chi$ be a primitive Dirichlet character modulo $M$, which we assume to be prime for simplicity. We…

Let $\tau$ denote the divisor function, and $f$ be any multiplicative function that satisfies some mild hypotheses. We establish the asymptotic formula or non-trivial upper bound for the shifted convolution sum $\sum_{n \leq…

Ingham studied two types of convolution sums of the divisor function, namely the shifted convolution sum $\sum_{n \le N} d(n) d(n+h)$ and the additive convolution sum $\sum_{n < N} d(n) d(N-n)$ for integers $N, h$ and derived their…

We establish upper bounds for shifted moments of modular $L$-functions to a fixed modulus as well as quadratic twists of modular $L$-functions under the generalized Riemann hypothesis. Our results are then used to establish bounds for…

We establish generalized Gaussian bounds and local limit theorems with Gaussian-type error for the convolution powers of certain complex-valued functions on $\mathbb{Z}^d$. These global space-times estimates/error, which are sharp in…

We produce nontrivial asymptotic estimates for shifted sums of the form $\sum a(h)b(m)c(2m-h)$, in which $a(n),b(n),c(n)$ are un-normalized Fourier coefficients of holomorphic cusp forms. These results are unconditional, but we demonstrate…

Let $A_{\pi}(n,1)$ be the $(n,1)$-th Fourier coefficient of the Hecke-Maass cusp form $\pi$ for $\rm SL_3(\mathbb{Z})$ and $ \omega(x)$ be a smooth compactly supported function. In this paper, we prove a nontrivial upper bound for the sum…

Given a cuspidal automorphic form $\pi$ on $\GL_2$, we study smoothed sums of the form $\sum_{n\in\mathbb{N}} a_{\pi}(n^2+d)W(\frac{n}{Y})$. The error term we get is sharp in that it is uniform in both $d$ and $Y$ and depends directly on…