I have attended my first course in analytic number theory (undergrad). I have encountered sums like Gauss' sum, Ramanujan sum and Kloosterman sum. My professor said that they are all over the place in analytic number theory. However, I only recognized that Gauss sum is applied to prove quadratic reciprocity law and some results about quadratic excess, nothing more.
Can somebody suggest more application of these kinds of sums in number theory?
Also, I rarely encounter books talking about Kloosterman sum. Can somebody suggest some good reference on this (perhaps gentle introduction; after all I am just undergrad!!!)? Thank you very much!
Suppose $X,Y \subseteq \{1,...,p-1\}$ for some prime $p$ and an integer $a$ which is coprime to $p$ you want to bound $$H = \#\{(x,y) \in X \times Y~:~ xy \equiv a \pmod{p} \}.$$
That is, you want to count points on a modular hyperbola. This shows up very naturally, for example, when dealing with the divisor function $\tau$. Then by carefully using orthogonality of characters we can write \begin{align*} H &= \frac{1}{p^2}\sum_{1 \leq x < p}\sum_{w \in X}\sum_{z \in Y}\sum_{r,s \in \mathbb{Z}/p\mathbb{Z}}e_p(r(x-w)+s(ax^{-1}-z)). \end{align*} Now lets rearrange to get \begin{align*} \frac{1}{p^2}\sum_{r,s \in \mathbb{Z}/p\mathbb{Z}}\sum_{1 \leq x <p}e_p(rx + sax^{-1})\sum_{w \in X}\sum_{z \in Y}e_m(-rw-sz). \end{align*} Now notice that $$\sum_{1 \leq x <p}e_p(rx + sax^{-1}) $$ is a Kloosterman sum. Using good bounds for this, you can get good bounds for the entire expression.