The Kloosterman sum is one of various trigonometric sums that are useful in number theory and, more generally, in finite harmonic analysis. The original Kloosterman sum is $$K_p(a,b)=\sum_{x\in\Fpstar} \exp \left( \frac{2\pi i(ax+bx^{-1})}{p} \right) $$ where $\Fp$ is the field of prime order $p$ . Such sums have been generalized in a few different ways since their introduction in 1926. For instance, let $q$ be a prime power, $\Fq$ the field of $q$ elements, $\chi:\Fqstar\to\C$ a character, and $\psi:\Fq\to\C$ a mapping such that $\psi(x+y)=\psi(x)\psi(y)$ identically. The sums $$K_\psi(\chi|a,b)=\sum_{x\in\Fqstar}\chi(x)\psi(ax+bx^{-1})$$ are of interest, because they come up as Fourier coefficients of modular forms.

Kloosterman sums are finite analogs of the $K$ -Bessel functions of this kind: $$K_s(a)=\frac{1}{2} \int_0^\infty x^{s-1}\exp\left(\frac{-a(x+x^{-1})}{2}\right) dx$$ where $\Re(a)>0$ .