Lehmer’s nonvanishing conjecture for Ramanujan’s tau-function

Establish that Ramanujan’s tau-function τ(n), the nth Fourier coefficient of the discriminant modular form Δ(z), is nonzero for all integers n ≥ 1, thereby proving Lehmer’s conjecture on the nonvanishing of τ(n).

Background

Lehmer’s conjecture asserts that the Ramanujan tau-function never vanishes. It remains a central open problem in the theory of modular forms and has motivated numerous studies on the arithmetic of τ(n), including questions about which integers and, in particular, which primes occur in its image.

Within this paper, the conjecture appears as part of the broader context on the distribution of values of τ(n). While the main results of the paper concern prime values of τ(n) under the abc Conjecture, the nonvanishing problem is a foundational backdrop that connects to the sparsity and structure of τ-values.