Toy models for D. H. Lehmer's conjecture

TL;DR

This paper explores simplified models related to Lehmer's conjecture, demonstrating non-vanishing Fourier coefficients for specific lattice shells, and linking spherical design properties to the conjecture.

Contribution

It introduces toy models using $ ext{Z}^2$ and $A_2$ lattices to study Lehmer's conjecture through spherical design properties.

Findings

Fourier coefficients of weighted theta series do not vanish for certain lattice shells.

Shells of specific norms in the studied lattices are not spherical 5- or 7-designs.

The results support the non-vanishing nature of these coefficients in simplified models.

Abstract

In 1947, Lehmer conjectured that the Ramanujan $\tau$-function $\tau (m)$ never vanishes for all positive integers $m$, where the $\tau (m)$ are the Fourier coefficients of the cusp form $\Delta_{24}$ of weight 12. Lehmer verified the conjecture in 1947 for $m<214928639999$. In 1973, Serre verified up to $m<10^{15}$, and in 1999, Jordan and Kelly for $m<22689242781695999$. The theory of spherical $t$-design, and in particular those which are the shells of Euclidean lattices, is closely related to the theory of modular forms, as first shown by Venkov in 1984. In particular, Ramanujan's $\tau$-function gives the coefficients of a weighted theta series of the $E_{8}$-lattice. It is shown, by Venkov, de la Harpe, and Pache, that $\tau (m)=0$ is equivalent to the fact that the shell of norm $2m$ of the $E_{8}$-lattice is an 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design. Lehmer's conjecture is difficult to prove, and still remains open. In this paper, we consider toy models of Lehmer's conjecture. Namely, we show 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 5 (resp. 7)-design does not exist among the shells in the $\mathbb{Z}^2$-lattice (resp. $A_{2}$-lattice).