Katz's Sato–Tate conjecture for Kloosterman sums

Let $a$ be a non-zero integer. For each prime $p$, let $\theta_p(a)\in[0,\pi]$ be the Kloosterman sum angle defined by

$$p^{1/2}\operatorname{Kl}(a;p):=S(a,1;p)=2p^{1/2}\cos\theta_p(a).$$

Katz's Sato–Tate conjecture. For every $f\in\mathcal{C}([0,\pi])$,

$$\lim_{x\rightarrow+\infty}\frac{1}{\pi(x)}\sum_{p\leq x}f\left(\theta_p(a)\right)=\frac{2}{\pi}\int_0^\pi f(\theta)\sin^2\theta\,d\theta.$$

This conjecture asserts that the Kloosterman sum angles are equidistributed with the Sato–Tate measure. It is presented as the analogue of the Sato–Tate conjecture for elliptic curves and provides the expected distribution underlying sign-change questions for Kloosterman sums.