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](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1981__54_/PMIHES_1981__54__123_0/PMIHES_1981__54__123_0.pdf), 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.



Best results regarding the Lang-Trotter conjecture
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 $x\_n \rightarrow \infty$.


