  Let $\tau(n)$ be Ramanujan's tau function, defined by the discriminant
modular form \[ \Delta(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\
\sum_{n=1}^{\infty}\tau(n) q^n \,,q=e^{2\pi i z} \] (this is the unique
holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer's
conjecture asserts that $\tau(n)\neq 0$ for all $n\geq 1$; since $\tau(n)$ is
multiplicative, it suffices to study primes $p$ for which $\tau(p)$ might
possibly be zero. Assuming standard conjectures for the twisted symmetric power
$L$-functions associated to $\tau$ (including GRH), we prove that if $x\geq
10^{50}$, then \[ \#\{x < p\leq 2x: \tau(p) = 0\} \leq 1.22 \times 10^{-5}
\frac{x^{3/4}}{\sqrt{\log x}},\] 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.
