On summatory functions for automorphic $L$-functions

Let $\lambda_f(n)$ denote the $n$th normalized Fourier coefficient of a primitive holomorphic cusp form $f$ for the full modular group. Let $\Delta(x,f\otimes f)$ be the error term in the asymptotic formula of Rankin and Selberg for
$$ \sum_{n\le x}\lambda_f(n)^2. $$
It is proved that $\Delta(x,f\otimes f)=\Omega(x^{3/8})$ and
$$ \sum_{n\le x}\lambda_f(n^2)=\Omega(x^{1/3}). $$
Other summatory functions associated with automorphic $L$-functions are also studied.