A variant of Lehmer's conjecture

Lehmer's conjecture asserts that τ(p)≠0 where τ is the Ramanujan τ-function. This is equivalent to the assertion that τ(n)≠0 for any n. A related problem is to find the distribution of primes p for which . These are open problems. We show that the variant of estimating the number of integers n for which n and τ(n) do not have a non-trivial common factor is more amenable to study. In particular, we show that the number of such n⩽x is ≪x/logloglogx. We prove a similar result for more general cusp forms. This may be seen as a modular analogue of an old result of Erdős on the Euler ϕ function.