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Kloosterman's Sum

Kloosterman's sum is defined by

S(u,v,n)=sum_(h)exp[(2pii(uh+vh^_))/n],
(1)

where h runs through a complete set of residues relatively prime to n and h^_ is defined by

hh^_=1 (mod n).
(2)

The notation K_n(u,v) is also used, at least for prime n.

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 ax_1^2+bx_2^2+cx_3^2+dx_4^2. Weil improved on Kloosterman's estimate for Ramanujan's problem with the best possible estimate

|S(u,u,n)|<=2sqrt(n)
(4)

(Duke 1997).

See also

Gaussian Sum

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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.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 SSSR 133, 1017-1020, 1960. English translation in Soviet Math. Dokl. 1, 928-932, 1960.Ramanujan, S. "On the Expression of a Number in the Form ax^2+by^2+cz^2+du^2." In Collected Papers of Srinivasa Ramanujan. (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: Amer. Math. Soc., 2000.

Referenced on Wolfram|Alpha

Kloosterman's Sum

Cite this as:

Weisstein, Eric W. "Kloosterman's Sum." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KloostermansSum.html

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