TITLE: Bounds on Fourier coefficients for $GL(3)$
QUESTION [5 upvotes]: I am referring for instance to this question about coefficients of automorphic forms on $GL(3)$. I know that the Ramanujan on average bound is known and gives
$$\sum_{n^2 m < x} |\lambda(n,m)|^2 \ll x^{1+\varepsilon}.$$
Is there anything known (in terms of upper bounds, with explicit dependence in the fixed $m$ or $n$) about the partial sums (except trivially bounding by the above) :
$$\sum_{n^2< x} |\lambda(n,m)|^2 \qquad \text{and} \qquad \sum_{m < x} |\lambda(n,m)|^2 ?$$

REPLY [2 votes]: On the Ramanujan conjecture $\lambda(m,n) \ll 1$. As there is no cancellation in the second sum, essentially (upto $x^\epsilon$) the best upper bound which one may expect for that is $x^{1+\epsilon}$.
For the first sum for the same reason the best possible bound would be $x^{1/2+\epsilon}$. To prove that note that $\lambda(n,m)=\overline{\lambda(m,n)}$. So 
$$\sum_{n^2<x}|\lambda(n,m)|^2=\sum_{n<\sqrt{x}}|\lambda(m,n)|^2\ll x^{1/2+\epsilon},$$
using the bound from the second sum.