In 1947, Lehmer conjectured that the Ramanujan τ-function τ(
m) never vanishes for all positive integers
m, where τ(
m) are the Fourier coefficients of the cusp form Δ
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 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 2
m 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
Z2-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
Z2-lattice (resp.
A2-lattice).
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