I am trying to understand [Kloosterman](https://en.wikipedia.org/wiki/Kloosterman_sum) sums and their estimates (e.g. from [[1](https://people.math.ethz.ch/~kowalski/exponential-sums-icms.pdf)], which does not prove)


$$ \Big| S(m,n;c) \Big| = \Big| \sum\_{(x,c) = 1} e\big( \frac{mx + nx^{-1}}{c}\big) \Big| < 2\, c^{3/4} $$


Another resource [[2](http://www.math.ucla.edu/~i707107/WeilBound-CharacterTwistedSum.pdf)] has a different version of the Kloosterman sum bound:


$$ \bigg| \sum\_{x \in \mathbb{F}\_q^\ast} \chi(x) \, e \big( \frac{ax + bx^{-1}}{q} \big) \bigg| \leq 2 \sqrt{q} $$


Yet another place has that if $m= n$ a different bound occurs. Perhaps I'm mixing up notations:


$$ K(n,n; c) \leq (n,c)^{1/2} \sqrt{c} \, \tau(c) $$


where $\tau(c)$ is the number of divisors of $c$ (and $K$ is just the same as $S$). This could be a refinement of the first two.




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The proofs of these results are very complicated. Is there a group action involed? Is there an interpretation via group theory of this inequality? Looks like there could be a hyperbola involved and that behaves well under fractional linear transformations like $x \mapsto x + 1$ or $x \mapsto - \frac{1}{x}$ in $\mathbb{F}\_q^\ast$.



group theory behind the Kloosterman bound $| S(m,n;c) |< 2\, c^{3/4}$
You can find a concise but self-contained proof of Weil's bound for Kloosterman sums of prime moduli [in these notes](http://www.renyi.hu/~gharcos/weil.pdf). Note that prime moduli constitute the really hard case.


