Toy models for D. H. Lehmer's conjecture II

TL;DR

This paper extends previous work on toy models related to Lehmer's conjecture by analyzing Fourier coefficients of weighted theta series for certain 2D lattices, showing non-vanishing for nonempty shells, implying the nonexistence of specific spherical designs.

Contribution

It generalizes prior results to lattices from imaginary quadratic fields with class number 1 or 2, excluding two special cases, demonstrating non-vanishing Fourier coefficients and the absence of certain spherical 2-designs.

Findings

Fourier coefficients do not vanish for nonempty shells in these lattices.

Spherical 2-designs do not exist among the nonempty shells.

Results extend previous models to broader classes of lattices.

Abstract

In the previous paper, we studied the "Toy models for D. H. Lehmer's conjecture". Namely, 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.