Conformal designs and D.H. Lehmerʼs conjecture

In 1947, Lehmer conjectured that the Ramanujan τ-function τ(m) is non-vanishing for all positive integers m, where τ(m) are the Fourier coefficients of the cusp form Δ of weight 12. It is known that Lehmerʼs conjecture can be reformulated in terms of spherical t-design, by the result of Venkov. In this paper, we show that τ(m)=0 is equivalent to the fact that the homogeneous space of the moonshine vertex operator algebra (V♮)m+1 is a conformal 12-design. Therefore, Lehmerʼs conjecture is now reformulated in terms of conformal t-designs.