Kloosterman's Sum

$$S(u,v,n)\equiv \sum_n \mathop{\rm exp}\nolimits \left[{2\pi i(uh+v\bar h)\over n}\right],$$ (1)

where $h$ runs through a complete set of residues Relatively Prime to $n$, and $\bar h$ is defined by

$$h\bar h\equiv 1\ \left({{\rm mod\ } {n}}\right).$$ (2)

If $(n,n')=1$ (if $n$ and $n'$ are Relatively Prime), then

$$S(u,v,n)S(u,v',n')=S(u,vn'^2+v'n^2,nn').$$ (3)

Kloosterman's sum essentially solves the problem introduced by Ramanujan of representing sufficiently large numbers by Quadratic Forms $a{x_1}^2+b{x_2}^2+c{x_3}^2+d{x_4}^2$. Weil improved on Kloosterman's estimate for Ramanujan's problem with the best possible estimate

$$\vert S(u,u,n)\vert\leq 2\sqrt{n}$$ (4)

(Duke 1997).

See also Gaussian Sum

References

Duke, W. ``Some Old Problems and New Results about Quadratic Forms.'' Not. Amer. Math. Soc. 44, 190-196, 1997.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 56, 1979.

Katz, N. M. Gauss Sums, Kloosterman Sums, and Monodromy Groups. Princeton, NJ: Princeton University Press, 1987.

Kloosterman, H. D. ``On the Representation of Numbers in the Form $ax^2+by^2+cz^2+dt^2$.'' Acta Math. 49, 407-464, 1926.

Ramanujan, S. ``On the Expression of a Number in the Form $ax^2+by^2+cz^2+du^2$.'' Collected Papers. New York: Chelsea, 1962.