Gaussian Sum

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.

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.