Refined Lang–Trotter conjecture for varying Frobenius traces

Let $E$ be an elliptic curve over $\mathbb{Q}$), let $C>1$, and let $r$ range uniformly over integers satisfying $|r|\leq 2\sqrt{x}$; exclude $r=0$ when $E$ has complex multiplication. Define

$$\Phi_E(z)=\begin{cases}\sqrt{1-z^2},&\text{if $E$ does not have CM},\\dfrac{1}{\sqrt{1-z^2}},&\text{if $E$ has CM}. \end{cases}$$

Refined Lang–Trotter conjecture. Uniformly in this range,

$$\pi_{E,r}(x)=C_{E,r}\int_{\max\{2,r^2/4\}}^x\frac{\Phi_E(r/(2\sqrt{t}))}{2\sqrt{t}\log t}\,dt+O_{E,C}\left(\frac{\sqrt{x}}{(\log x)^C}\right),$$

where $C_{E,r}$ is the constant in the Lang–Trotter conjecture. This refinement incorporates the Sato–Tate or CM distribution and the fact that primes counted must satisfy $p\geq r^2/4$; its uniform validity remains open.