arXiv subjects
Publications and source records attributed to Himanshi Chanana.
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.
Let $A(n)$ be the $(1,n)$-th Fourier coefficients of $SL(3,\mathbb{Z})$ Hecke-Maass cusp form, denoted as $A(1,n)$ or the triple divisor function, denoted as $d_3(n)$. Let $k \geqslant3$ be an integer. In this paper, we establish an asymptotic formula for the sum \begin{equation*} \mathop{\sum}_{\substack{1 \leqslant n_1, n_2 \leqslant X^{1/2} \\ 1 \leqslant n_3 \leqslant X^{1/k}}} A(Q(n_1,n_2) + n_3^k)\mathsf{a}(n_3), \end{equation*} where $\mathsf{a}(n)$ is either von-Mangoldt function or identity function, and $Q(x,y) \in \mathbb{Z}[x,y]$ is a binary quadratic polynomial. When $A(n)=A(1,n)$, then $\mathsf{a}(n)$ can be any bounded arithmetical function.
Let $A(n)$ denote the $(1,n)\text{-th}$ Fourier coefficient of a $SL(3, \mathbb{Z})$ Hecke eigenform or the ternary divisor function $d_3(n)$. Let $Q(x,y)$ be a symmetric positive definite quadratic form. This article establishes an asymptotic formula with a power-saving error term for the following sum \begin{equation*} \sum_{1 \leqslant m \leqslant X} \sum_{1 \leqslant n\leqslant Y} A(Q(m,n)), \end{equation*} where $X>1$ and $Y\leqslant X$.