Kloosterman sum

The Kloosterman sum is one of various trigonometric sums that areuseful in number theory and, more generally, in finite harmonicanalysis.The original Kloosterman sum is

$$K_p(a,b)=\sum _{x\in {𝔽}_p^{*}}\exp \left(\frac{2\pi i(ax+b{x}^{-1})}{p}\right)$$

where ${𝔽}_p$ is the field of prime order $p$.Such sums have been generalizedin a few different ways since their introduction in 1926.For instance, let $q$ be a prime power, ${𝔽}_q$ the fieldof $q$ elements, $\chi :{𝔽}_q^{*}\to ℂ$ a character, and$\psi :{𝔽}_q\to ℂ$a mapping such that $\psi (x+y)=\psi (x)\psi (y)$ identically.The sums

$$K_{\psi }(\chi |a,b)=\sum _{x\in {𝔽}_q^{*}}\chi (x)\psi (ax+b{x}^{-1})$$

are of interest, because they come up as Fourier coefficientsof modular forms.

Kloosterman sums are finite analogs of the $K$-Besselfunctions of this kind:

$$K_s(a)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}{x}^{s-1}\exp \left(\frac{-a(x+x^{-1})}{2}\right)𝑑x$$

where $\mathrm{\Re }(a)>0$.