Sumit Kumar: On the Rankin-Selberg bounds

Given a Hecke holomorphic/Maass cusp form $f$ for $SL(2, \mathbb{Z})$, the  classical Rankin-Selberg method yields the following bound
$$\sum_{n \leq X} \lambda_f(n)^2= L(1,\mathrm{ sym}^2 f)X+O(X^{3/5}),$$
for any $X \geq 1$.
In this talk, we discuss how to improve on the error term. We prove the following bound
$$\sum_{n \leq X} \lambda_f(n)^2= L(1,\mathrm{ sym}^2 f)X+O(X^{3/5-3/205+\epsilon}),$$
for any $\epsilon>0$. This improves on the recent bound $O(X^{3/5-3/305+\epsilon})$ by Huang. This is a joint work with P. Sharma.