The explicit Sato-Tate conjecture and densities pertaining to Lehmer-type questions

Let (Formula Presented) be a newform with squarefree level N that does not have complex multiplication. For a prime p, define θ_p ∈ [0, π] to be the angle for which a_f (p) = 2p^(k−1)/2 cos θ_p. Let I ⊂ [0, π] be a closed subinterval, and let (Formula Presented) be the Sato-Tate measure of I. Assuming that the symmetric power L-functions of f satisfy certain analytic properties (all of which follow from Langlands functoriality and the Generalized Riemann Hypothesis), we prove that if x is sufficiently large, then (Formula Presented) with an implied constant of 3.33. By letting I be a short interval centered at π/2 and counting the primes using a smooth cutoff, we compute a lower bound for the density of positive integers n for which af (n) ≠ 0. In particular, if τ is the Ramanujan tau function, then under the aforementioned hypotheses, we prove that (Formula Presented). We also discuss the connection between the density of positive integers n for which af (n) ≠ 0 and the number of representations of n by certain positivedefinite, integer-valued quadratic forms.