arXiv subjects
Publications and source records attributed to Mohd Harun.
This article proves non-trivial estimates for a bilinear sum involving the Fourier coefficients of a Hecke-holomorphic or Hecke-Maass cusp form for $\mathrm{SL}(2,\mathbb{Z})$. As corollaries, we draw interesting results related to non-trivial bounds of different shifted convolution sums and summatory functions.
This article will prove non-trivial estimates for the average and weighted average version of general $GL(3) \times GL(3)$ shifted convolution sums by using the circle method.
\begin{abstract} In this article, we will get non-trivial estimates for the central values of degree six Rankin-Selberg $L$-functions $L(1/2+it, π\times f)$ associated with a ${GL(3)}$ form $π$ and a ${GL(2)} $ form $f$ using the delta symbol approach in the hybrid settings i.e. in the level of ${GL(2)}$ form and $t$-aspect. \end{abstract}
In this paper, we will prove the non-trivial bound for the weighted average version of shifted convolution sum for $GL(3)\times GL(2)$, i.e. for any $ε>0$ and $X^{1/4+δ} \leq H \leq X$ with $δ>0$, \[ \frac{1}{H}\sum_{h=1}^\infty λ_f(h) V\left( \frac{h}{H}\right)\sum_{n=1}^\infty λ_π(1,n) λ_g (n+h) W\left( \frac{n}{X} \right)\ll X^{1-δ+ε} \] where $V,W$ are smooth compactly supported funtions, $λ_f(n), λ_g(n)$ and $λ_π(1,n)$ are the normalized n-th Fourier coefficients of $SL(2,\mathbb{Z})$ Hecke-Maass cusp forms $f,g$ and $SL(3,\mathbb{Z})$ Hecke-Maass cusp form $π$, respectively.