In https://arxiv.org/abs/1009.3956 it is stated that there is a $c>0$ such that $\pi(x)\bmod2$ can be computed in time $O(x^{\frac12-c})$ (more precisely number of primes $\bmod 2$ in $[x,2x]$ can be computed in time $O(x^{\frac12-c})$).
Is there a good estimate known for $c$ and is $\frac12-c=o(1)$ believed achievable?
Is the technique applicable to union of disjoint intervals without testing them separately?