Conformal designs and D.H. Lehmer's conjecture

Abstract

In 1947, Lehmer conjectured that the Ramanujan \tau-function \tau(m) is non-vanishing for all positive integers m, where \tau(m) are the Fourier coefficients of the cusp form \Delta of weight 12. It is known that Lehmer's conjecture can be reformulated in terms of spherical t-design, by the result of Venkov. In this paper, we show that \tau(m) = 0 is equivalent to the fact that the homogeneous space of the moonshine vertex operator algebra (V^\natural)_{m+1} is a conformal 12-design. Therefore, Lehmer's conjecture is now reformulated in terms of conformal t-designs.