Sumit Kumar: Delta method and its application to the Rankin-Selberg problem

Abstract:

In this talk we consider the bilinear sum \(\begin{equation} B(L, N/L ) = \sum_{\ell \leq L}\sum_{m\ell \leq N} A_\pi (1,m) e(-c(\ell m )^{\beta})V(m\ell/N), \end{equation}\) where \(A_\pi (1,m)\) are the normalised Fourier coefficients of an \(SL_3(\mathbb{Z})\).

Hecke Maass cusp form \(\pi\) and discuss on how to get non-trivial cancellations in \(B(L,N/L)\). As an application we improve upon the known error term bound (due to Huang) in the Rankin-Selberg problem, i.e., we prove the following bound \(\begin{equation}\sum_{n \leq X} \lambda_f(n)^2= L(1,\mathrm{ sym}^2 f)X+O_f(X^{3/5-3/205+\epsilon}),\end{equation}\) where \(f\) is a Hecke holomorphic/Maass cusp form for \(SL(2, \mathbb{Z})\).

Ongoing joint work with P. Sharma.