The Explicit Sato-tate Conjecture and Densities Pertaining to Lehmer-type Questions

Let f(z) = ∑∞ n=1 af (n)q n ∈ S k (Γ0(N)) be a newform with squarefree level N that does not have complex multiplication. For a prime p, define θp ∈ [0, π] to be the angle for which af (p) = 2p (k−1)/2 cos θp. Let I ⊂ [0, π] be a closed subinterval, and let dμST = 2 π sin 2 θdθ 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 ∣∣∣∣#{p ≤ x : θp ∈ I} − μST (I)∫ x 2 dt log t ∣∣∣∣ x log(Nkx) log x with an implied constant of 3.33. By letting I be a short interval centered at π2 and counting the primes using a smooth cutoff, we compute a lower bound for the density of positive integers n for which af (n) 6= 0. In particular, if τ is the Ramanujan tau function, then under the aforementioned hypotheses, we prove that lim x→∞ #{n ≤ x : τ(n) 6= 0} x > 1− 1.54× 10−13. We also discuss the connection between the density of positive integers n for which af (n) 6= 0 and the number of representations of n by certain positive-definite, integer-valued quadratic forms.