Toy Models for D. H. Lehmer’s Conjecture

In 1947, Lehmer conjectured that the Ramanujan τ -function τ(m) never vanishes for all positive integers m, where the τ(m) are the Fourier coefficients of the cusp form ∆24 of weight 12. Lehmer verified the conjecture in 1947 for m < 214928639999. In 1973, Serre verified up to m < 10, 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 τ -function gives the coefficients of a weighted theta series of the E8-lattice. It is shown, by Venkov, de la Harpe, and Pache, that τ(m) = 0 is equivalent to the fact that the shell of norm 2m of the E8-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 Z-lattice and the A2-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 Z-lattice (resp. A2-lattice).