Leírás
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, which is a follow up of the previous talk, we continue our analysis to prove the following result
\[\sum_{n \leq X} \lambda_f(n)^2= L(1,\mathrm{ sym}^2 f)X+O(X^{3/5-3/205+\epsilon}),\]
for any $\epsilon>0$. In particular, we show how to reduce the problem to getting non-trivial cancellations in the following bilinear exponential sum
\[ B(T, X):= \sum_{\ell \ll N}\sum_{m \sim N/\ell} \lambda_{\mathrm{ sym}^2 f}(m) e(-4(\ell mX )^{1/4})V(m\ell/N), \]
where $N \sim T^4/X$, and $T \ll X/Y$, with $Y < X^{1-\epsilon}$, a parameter.