Conformal designs and D.H. Lehmer's conjecture

TL;DR

This paper links Lehmer's conjecture on the non-vanishing of Ramanujan's tau-function to the concept of conformal t-designs within moonshine vertex operator algebras, providing a new reformulation of the conjecture.

Contribution

It establishes an equivalence between tau(m)=0 and the conformal 12-design property of a homogeneous space in moonshine VOA, offering a novel perspective on Lehmer's conjecture.

Findings

Tau(m)=0 iff the space is a conformal 12-design

Reformulation of Lehmer's conjecture in terms of conformal t-designs

Connects number theory with vertex operator algebra theory

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.