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Cubic Gauss Sum

A cubic Gauss sum is a character sum combining a multiplicative character of order 3 with a nontrivial additive character of a finite field. For Eisenstein integers O=Z[omega], with omega=e^(2pii/3), a primary prime pi satisfies pi=1 (mod3). Let chi_pi be its cubic residue character, extended by zero at zero, and let N(z)=zz^_ and Tr(z)=z+z^_. The normalized sum is

G(pi)=1/(sqrt(N(pi)))sum_(vmodpi)chi_pi(v)e^(2piiTr(v/pi)).

For nonzero residues, chi_pi(v) is the cube root of unity congruent to v^((N(pi)-1)/3) modulo pi (Dunn and Radziwiłł 2024). The complex conjugate of G(pi) occurs, up to a constant factor, as a Fourier coefficient of Kubota's cubic theta function (Dunn and Radziwiłł 2024).

See also

Eisenstein Integer, Gamma Function, Gauss Sum, Kubota's Cubic Theta Function, Multiplicative Character, Prime Number

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References

Dunn, A. and Radziwiłł, M. "Bias in Cubic Gauss Sums: Patterson's Conjecture." Ann. Math. 200, 967-1057, 2024. https://doi.org/10.4007/annals.2024.200.3.3.

Cite this as:

Weisstein, Eric W. "Cubic Gauss Sum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CubicGaussSum.html

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