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
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.Kloosterman, H. D. "The
Behavior of General Theta Functions under the Modular Group and the Characters of
Binary Modular Congruence Groups, I." Ann. Math.47, 317-375,
1946.Kloosterman, H. D. "The Behavior of General Theta Functions
under the Modular Group and the Characters of Binary Modular Congruence Groups, II."
Ann. Math.47, 376-447, 1946.Malyšev, A. V.
"Gauss and Kloosterman Sums." Dokl. Akad. Nauk SSSR133,
1017-1020, 1960. English translation in Soviet Math. Dokl.1, 928-932,
1960.Ramanujan, S. "On the Expression of a Number in the Form ." In Collected
Papers of Srinivasa Ramanujan. (Ed. G. H. Hardy, P. V. S. Aiyar,
and B. M. Wilson). Providence, RI: Amer. Math. Soc., 2000.