A variant of Lehmer's conjecture

Abstract: Lehmer''s conjecture asserts that where τ is the Ramanujan τ-function. This is equivalent to the assertion that 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 do not have a non-trivial common factor is more amenable to study. In particular, we show that the number of such is . 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. [Copyright &y& Elsevier]