On the Cubic Theta Function

Abstract

The generalized theta function of a totally imaginary field including n-th roots of unity, which was defined by T. Kubota [2], was introduced in his investigation of the reciprosity law of the n-th power residue. If n = 2, it reduces to the classical theta function. In the case n = 3 for the Eisenstein field, the Fourier coefficients of the cubic theta function, which were explicitly expressed by S.J. Patterson, are essentially cubic Gauss sums [3], Furthermore in the case n = 4 for the Gaussian field those of the biquadratic theta functions, which have been investigated by T. Suzuki [4], haven’t been obtained completely yet.