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

Let $f$ be a holomorphic or Maass cusp forms for $ \rm SL_2(\mathbb{Z})$ with normalized Fourier coefficients $\lambda_f(n)$ and \bna r_{\ell}(n)=\#\left\{(n_1,\cdots,n_{\ell})\in \mathbb{Z}^2:n_1^2+\cdots+n_{\ell}^2=n\right\}. \ena Let…

Let $g$ be a fixed holomorphic cusp form of arbitrary level and nebentypus. Let $\chi$ be a primitive character of prime-power modulus $q = p^{\gamma}$. In this paper, we prove the following hybrid Weyl-type subconvexity bound…

Let $X$ be a regular linear diffusion whose state space is an open interval $E\subseteq\mathbb{R}$. We consider a diffusion $X^*$ which probability law is obtained as a Doob $h$-transform of the law of $X$, where $h$ is a positive harmonic…

For $m\ge 2$, let $\pi$ be an irreducible cuspidal automorphic representation of $GL_m(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. Let $a_\pi(n)$ be the $n^{th}$ coefficient of the $L$-function attached to $\pi$. Goldfeld and…

We calculate mean values of $\operatorname{GL}_n$-automorphic $L$-functions twisted by primitive even Dirichlet characters of prime-power conductor, at arbitrary points within the critical strip, by derivation of special Voronoi summation…

Consider a measure $\mu_\lambda = \sum_x \xi_x \delta_x$ where the sum is over points $x$ of a Poisson point process of intensity $\lambda$ on a bounded region in $d$-space, and $\xi_x$ is a functional determined by the Poisson points near…

Let $\{\lambda_f(n)\}_{n \geq 1}$ be the normalized Hecke eigenvalues of a given holomorphic cusp form $f$ of even weight $k$. We show under the assumption of the existence of Littlewood's type zero free region for $L(s, f, \chi)$, where…

Distorted sums of models were introduced and discussed in [Sh:463]. This notion generalizes the notion of disjoint (or direct) sums of models by letting the summands overlap. In the first section we investigate types in distorted sums and…

We derive an analytic model for the redshift evolution of linear-bias, allowing for interactions and merging of the mass-tracers, by solving a second order differential equation based on linear perturbation theory and the Friedmann-Lemaitre…

Let $f\in S_k(N,\psi)$ be a newform, and let $\chi$ be a primitive character of conductor $q^{\ell}$. Assume that $q$ is a prime and $\ell>1$. In this paper we describe a method to establish convexity breaking bounds of the form $$…

We apply three forward light-by-light scattering sum rules to charmonium states. We show that these sum rules imply a cancellation between charmonium bound state contributions, which are mostly known from the $\gamma \gamma$ decay widths of…

We introduce a shifted convolution sum that is parametrized by the squarefree natural number $t$. The asymptotic growth of this series depends explicitly on whether or not $t$ is a \emph{congruent number}, an integer that is the area of a…

We prove uniform estimates for the decay rate of the Fourier transform of measures supported on real-analytic hypersurfaces in R^3. If the surface contains the origin and is oriented such that its normal at the origin is in the direction of…

Let $f$ be a full-level cusp form for $GL_m(\mathbb Z)$ with Fourier coefficients $A_f(n_1,...,n_{m-1})$. In this paper an asymptotic expansion of Voronoi's summation formula for $f$ is established. As applications of this formula, a…

In this paper, we investigate a weighted divisor problem involving the exponential sum of $D_{(1)}(n)$, the $n$th coefficient in the Dirichlet series expansion of $\zeta'(s)^2$. We establish a truncated Vorono\"{i} type formula for the…

We continue our study of convolution sums of two arithmetical functions $f$ and $g$, of the form $\sum_{n \le N} f(n) g(n+h)$, in the context of heuristic asymptotic formul\ae. Here, the integer $h\ge 0$ is called, as usual, the {\it shift}…

We study the vector space V_k^m(\lambda) of shifted polyharmonic Maass forms of weight k \in 2Z, depth m \geq 0, and shift \lambda \in C. This space is composed of real-analytic modular forms of weight k for PSL(2,Z) with moderate growth at…

Let $\lambda$ denote the Liouville function. A well known conjecture of Chowla asserts that for any distinct natural numbers $h_1,\dots,h_k$, one has $\sum_{1 \leq n \leq X} \lambda(n+h_1) \dotsm \lambda(n+h_k) = o(X)$ as $X \to \infty$.…

The expansion of a square integrable function on $SL(2,C)$ into the sum of the principal series matrix coefficients with the specially selected representation parameters was recently used in the Loop Quantum Gravity $\cite{RovelliBook2}$,…

We obtain a first moment formula for Rankin-Selberg convolution $L$-series of holomorphic modular forms or Maass forms of arbitrary level on $GL(2)$, with an orthonormal basis of Maass forms. One consequence is the best result to date,…