A Gaussian sum is a sum of the form

(1)

where
and are relatively
prime integers . The symbol is sometimes used instead of . 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 (Borwein and Borwein 1987, pp. 83 and 86).

If , then

(2)

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

(3)

for odd . Written explicitly

(4)

(Nagell 1951, p. 177).

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

(5)

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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References 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. Referenced on Wolfram|Alpha Gaussian Sum
Cite this as:
Weisstein, Eric W. "Gaussian Sum." From
MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/GaussianSum.html

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