The explicit Sato-Tate conjecture for primes in arithmetic progressions

Let τ(n) be Ramanujan's tau function, defined by the discriminant modular form Δ(z) = q-j=1∞(1 - qj)24 =ζ n=1∞τ(n)qn,q = e2πiz (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer's conjecture asserts that τ(n)=0 for all n ≥ 1; since τ(n) is multiplicative, it suffices to study primes p for which τ(p) might possibly be zero. Assuming standard conjectures for the twisted symmetric power L-functions associated to τ (including GRH), we prove that if x ≥ 1050, then #{x < p ≤ 2x:τ(p) = 0}≤ 1.22 × 10-5 x3/4 log x, a substantial improvement on the implied constant in previous work. To achieve this, under the same hypotheses, we prove an explicit version of the Sato-Tate conjecture for primes in arithmetic progressions.