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Erschienen in:
2013 | OriginalPaper | Buchkapitel
Toy Models for D. H. Lehmer’s Conjecture II
verfasst von : Eiichi Bannai, Tsuyoshi Miezaki
Erschienen in: Quadratic and Higher Degree Forms
Verlag: Springer New York
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Abstract
In the previous paper under the same title, we showed that the m-th Fourier coefficient of the weighted theta series of the \({\mathbb{Z}}^{2}\)-lattice and the A
2-lattice does not vanish when the shell of norm m of those lattices is not the empty set. In other words, the spherical 4 (resp. 6)-design does not exist among the nonempty shells in the \({\mathbb{Z}}^{2}\)-lattice (resp. A
2-lattice). This paper is the sequel to the previous paper. We take 2-dimensional lattices associated to the algebraic integers of imaginary quadratic fields whose class number is either 1 or 2, except for \(\mathbb{Q}(\sqrt{-1})\) and \(\mathbb{Q}(\sqrt{-3})\), then, show that the m-th Fourier coefficient of the weighted theta series of those lattices does not vanish, when the shell of norm m of those lattices is not the empty set. Equivalently, we show that the corresponding spherical 2-design does not exist among the nonempty shells in those lattices.