Let $E$ be an elliptic curve over $\mathbb Q$, which does not have complex multiplication over the algebraic closure of $\mathbb Q$. For $x>0$, let $P(x)$ be the number of primes $p < x$ such that $E$ has good **super-singular** reduction at $p$. The Lang-Trotter's conjecture states that
$$P(x) = O (x^{1/2}/\log x).$$
In Serre's paper Quelques applications du théorème de densité de Chebotarev, Serre
proves $P(x)=O(x^{3/4})$ under the generalized Riemann hypothesis (GRH) for Artin $L$-functions.

Has this bound been improved since? If so, what is the better known bound (under GRH) and where can I find it?

Thanks.