Gaussian Sum

A Gaussian sum is a sum of the form


where p and q are relatively prime integers. The symbol phi is sometimes used instead of S. Although the restriction to relatively prime integers is often useful, it is not necessary, and Gaussian sums can be written so as to be valid for all integer q (Borwein and Borwein 1987, pp. 83 and 86).

If (n,n^')=1, then


(Nagell 1951, p. 178). Gauss showed that


for odd q. Written explicitly

 S(-2,q)={(i+1)sqrt(q)   for q=0 (mod 4); sqrt(q)   for q=1 (mod 4); 0   for q=2 (mod 4); isqrt(q)   for q=3 (mod 4)

(Nagell 1951, p. 177).

For p and q of opposite parity (i.e., one is even and the other is odd), Schaar's identity states


Such sums are important in the theory of quadratic residues.

See also

Kloosterman's Sum, Quadratic Residue, Schaar's Identity, Singular Series

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Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Evans, R. and Berndt, B. "The Determination of Gauss Sums." Bull. Amer. Math. Soc. 5, 107-129, 1981.Katz, N. M. Gauss Sums, Kloosterman Sums, and Monodromy Groups. Princeton, NJ: Princeton University Press, 1987.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.Nagell, T. "The Gaussian Sums." §53 in Introduction to Number Theory. New York: Wiley, pp. 177-180, 1951.Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 132-134, 1994.

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Gaussian Sum

Cite this as:

Weisstein, Eric W. "Gaussian Sum." From MathWorld--A Wolfram Web Resource.

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