Does anybody know the non-trivial bound for this sum?
$S(m,n,c,q)=\sum_{a,b\in \mathbb{Z}/cq\mathbb{Z}, \;ad\equiv 1\text{ mod }c} e^{2\pi i(am+nd)/qc},$
where $m,n\in\mathbb{Z},\;q,c\in\mathbb{N}$.
This seems to be not general bound but I want to know if there is conditions for $q$ for the Weil like bound to be valid, i.e., $|S(m,n,c,q)|\ll O(c^{1/2})$. Thanks.
Setting $a=a_1+ca_2$, $d=d_1+cd_2$ we'll get $$S(m,n,c,q)=\sum_{a_1d_1\equiv 1\pmod c}e\left(\frac{a_1m+d_1n}{cq}\right)\sum_{a_2,d_2=1}^qe\left(\frac{a_2m+d_2n}{q}\right).$$ The inner (linear) sums can be calculated explicitely $$S(m,n,c,q)=q^2\delta_q(m)\delta_q(n)\sum_{a_1d_1\equiv 1\pmod c}e\left(\frac{a_1m/q+d_1n/q}{cq}\right)=q^2\delta_q(m)\delta_q(n)S(m/q,n/q,c),$$ where $S(m,n,c)$ is a usual Kloosterman sum and $\delta_q$ is a charaxteristic function of divisibility by $q$.
