Proof of the Lehmer conjecture on Ramanujan's $\tau$ function

A criterion for Lehmer’s conjecture in terms of the spherical designs held in the shells of the lattice $E_8$ was derived by de La Harpe, Pache and Venkov circa 2005. We check that this criterion is satisfied by combining spherical designs, harmonic polynomials, weighted theta series, and Deligne’s bound on the modulus of the $\tau$ function.