A Gaussian sum is a sum of the form S(p, q) congruent sum_(r = 0)^(q - 1) e^(-π i r^2 p/q), where p and q are relatively prime integers. The symbol φ 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. If (n, n') = 1, then S(m, n n') = S(m n', n) S(m n, n') (Nagell 1951, p. 178).

Kloosterman's sum | quadratic residue | Schaar's identity | singular series