Toy models for D. H. Lehmer's conjecture

Abstract

In 1947, Lehmer conjectured that the Ramanujan $\tau$-function $\tau (m)$ never vanishes for all positive integers $m$, where $\tau (m)$ 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 $\mathbf{Z}^2$-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 $\mathbf{Z}^2$-lattice (resp. $A_2$-lattice).