Kloosterman sum

In mathematics, a Kloosterman sum is a particular kind of exponential sum. They are named for the Dutch mathematician Hendrik Kloosterman, who introduced them in 1926^[1] when he adapted the Hardy–Littlewood circle method to tackle a problem involving positive definite diagonal quadratic forms in four variables, strengthening his 1924 dissertation research on five or more variables.^[2]

Let a, b, m be natural numbers. Then

${\displaystyle K(a,b;m)=\sum _{\stackrel {0\leq x\leq m-1}{\gcd(x,m)=1}}e^{{\frac {2\pi i}{m}}(ax+bx^{*})}.}$

Here x* is the inverse of x modulo m.