## Kloosterman's Sum

 (1)

where runs through a complete set of residues Relatively Prime to , and is defined by
 (2)

If (if and are Relatively Prime), then
 (3)

Kloosterman's sum essentially solves the problem introduced by Ramanujan of representing sufficiently large numbers by Quadratic Forms . Weil improved on Kloosterman's estimate for Ramanujan's problem with the best possible estimate
 (4)

(Duke 1997).

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 .'' Acta Math. 49, 407-464, 1926.

Ramanujan, S. On the Expression of a Number in the Form .'' Collected Papers. New York: Chelsea, 1962.