Sorry to disturb. Recently, I encountered a puzzle on the sums involving two Kloosterman sums. That is,
For any $h, q_1,q_2\in \mathbb{N}$ with $(q_1,q_2)=1$ and $Q>1$, how two get a bound
$$\sum_{h\sim Q}\, \sum_{q_1\sim Q} \, \sum_{q_2\sim Q} S(q_1+h,1;q_2)\,S(q_2+h,1;q_1)\ll Q^{4-\delta}$$ for some constant $\delta>0$? Here, $h\sim Q$ means $Q<h\le 2Q$.
One easily finds that an application of the Weil bound for Kloosterman sums yields the upper-bound of $O(Q^{4+\epsilon})$. Whether or not one could get better bound compared with this？
If any expert knows something concerning this question, please show some hints or some references. Much obliged.
Many many thanks in advance.
