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On the Cubic Theta Function
Published online by Cambridge University Press: 22 January 2016
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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.
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- Research Article
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- Copyright © Editorial Board of Nagoya Mathematical Journal 1987
References
[ 1 ]
Davenport, H. and Hasse, H., Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fallen, J. reine angew. Math., 172 (1934), 151–182.Google Scholar
[ 2 ]
Kubota, T., Some results concerning reciprosity law and real analytic automorphic functions, Proc. Sympos. Pure Math., 20 (1971), 382–395.Google Scholar
[ 3 ]
Patterson, S. J., A cubic analogue of the theta series, J. reine angew. Math., 296 (1977), 125–161.Google Scholar
[ 4 ]
Suzuki, T., Some results on the biquadratic theta series, J. reine angew Math., 340 (1984), 70–117.Google Scholar
[ 5 ]
Weil, A., Uber die Bestimmung Dirichletscher Reiehen durch Funktionalgleichungen, Math. Ann., 168 (1967), 149–156.Google Scholar
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