A Gaussian sum is a sum
of the form
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).
(Nagell 1951, p. 178). Gauss showed that
odd . Written explicitly
(Nagell 1951, p. 177).
and 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
See also Kloosterman's Sum
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References Borwein, J. M. and Borwein, P. B.
New York: Wiley, 1987. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. Evans, R. and Berndt, B. "The Determination
of Gauss Sums." Bull. Amer. Math. Soc. 5, 107-129, 1981. Katz,
N. M. Princeton, NJ: Princeton University
Press, 1987. Gauss
Sums, Kloosterman Sums, and Monodromy Groups. 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 New York: Wiley, pp. 177-180, 1951. Introduction
to Number Theory. Riesel,
H. Boston, MA: Birkhäuser,
pp. 132-134, 1994. Prime
Numbers and Computer Methods for Factorization, 2nd ed. Referenced on Wolfram|Alpha Gaussian Sum
Cite this as:
Weisstein, Eric W. "Gaussian Sum." From
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