Best results regarding the Lang-Trotter conjecture 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.
 A: A couple of years after my Ph.D. thesis (whose main result is the infinitude of
singular primes, i.e. $P(x) \rightarrow \infty$ as $x \rightarrow \infty$),
Kaneko published a result[1] that let me obtain the unconditional 
upper bound[2] $P(x) = O(x^{3/4} \log x)$ using some of the same ideas:

[1] Masanobu Kaneko: Supersingular $j$-invariants as singular moduli mod $p$, 
  Osaka J. Math. 26 (1989), 849–855.
[2] Noam D. Elkies: Distribution of supersingular primes, Astérisque 198-199-200 (1991; proceedings of Journées Arithmétiques 1989), 127–132.

As noted in [2], the factor $\log x$ can be removed with some
more care (averaging over the auxiliary discriminants $-D$ improves on
the worst-case estimate), thus exactly matching Serre's conditional bound.
As far as I know, no further improvement has been obtained since then,
even assuming GRH.
That paper [2] also reports lower bounds from my thesis, conditional on GRH
for quadratic characters: $P(x) \gg \log \log x$, and also
$P(x_n) \gg \log x_n$ for an infinite sequence $x_n \rightarrow \infty$.
