Part of this blog discussing the twin prime conjecture mentions a connection between three objects:
$ \sum_{x \leq n \leq 2x} \tau\Big(n(n+2)\Big) $ average over twin primes where $\tau(n) = (1 \ast 1)(n) = \sum_{d|n} 1$
$\# \Big\{ (a,b) : ab = -2 \mod d\Big\} \subseteq (\mathbb{Z}/d\mathbb{Z})^2$ by solving the equation $ab-cd=2$
$\sum_{m \in \mathbb{Z}/r\mathbb{Z}} e\Big( \frac{a_1 m + a_2 \overline{m}}{r} \Big)$ by Taking a Fourier transform
Item #1 is pretty clear. It is often equal to $\tau(n)\tau(n+2)$
That article glosses over numerous points.
How is #2 a statement about equidistribution over the hyperbola?
How is #3 the Fourier transform of item #2? What is the relation between $r$ and $d$?
Later on the blog proves the Kloosterman bound, some weak version of the Weil bound.
$$\sum_{m \in \mathbb{Z}/r\mathbb{Z}} e\Big( \frac{a_1 m + a_2 \overline{m}}{r} \Big) \ll r^{3/4 +o(1)}\mathrm{gcd}(a,b,r)$$
The article does a decent job of explaining that if we proove this bound the Kloosterman averages vanish (upon division by $r$).
Empirical Data Related to Equidistribution of $ab +2 = 0$ mod $d$
$d = 1500$
$d = 1000$
