Toy models for D. H. Lehmer's conjecture

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.