Let $f=\sum\_{n\ge 1}a\_nq^n$ be a normalized Hecke eigenform which is not of CM-type, of weight $k\ge 2$ for the congruence subgroup $\Gamma\_0(N)$. Let $a\in\mathbb{Z}$ and define


$$
\pi\_f(x,a):=\#\{p\le x\; :\; a\_p=a \}
$$
I am trying to understand the proof of Theorem 5.1 from K. Murty's paper "Modular Forms and
the Chebotarev Density Theorem II" which asserts that
$$\pi\_f(x,a)\ll \dfrac{x(\log\log x)^2}{\log^2x}\label{1}.\tag{1}$$
Using the Cauchy–Schwarz and Polya–Vinogradov inequalities, Murty proves that
$$\pi\_f(x,a)\ll \max\_{\ell\in I}\pi\_f(x,a,\ell)+O\left(\frac{\pi\_f(x,a)^{1/2}x^{1/2}}{\pi(I)^{1/2}}\right)\label{2},\tag{2}$$
where $I=[y,y+u]$. He then claims that \eqref{1} follows easily from \eqref{2} if we prove that
$$
\pi\_f(x,a,\ell)\ll \dfrac{x(\log\log x)^2}{\log^2x}.
$$
My question is: how can \eqref{1} be derived from \eqref{2}?



A question on a paper of K. Murty
There is a flaw in Murty's argument. Once one corrects this flaw, the bound \eqref{1} weakens to $\ll x(\log\log x)^3/(\log x)^2$. Fortunately, [Thorner and Zaman - A Chebotarev variant of the Brun–Titchmarsh theorem and bounds for the Lang–Trotter conjectures](https://arxiv.org/abs/1606.09238) (Sections 1 and 9) fixes the argument (using an upper bound for the Chebotarev prime counting function that is far more efficient than what Murty uses in his proof). In the end, Murty's claimed bound $\ll x(\log\log x)^2/(\log x)^2$ is still true (but with a lot more work). The nature of the flaw is described in Section 9. If you want to see a version of these results where the implied constants are computed, see [Hu, Iyer, and Shashkov - Modular forms and an explicit Chebotarev variant of the Brun–Titchmarsh theorem](https://arxiv.org/abs/2208.10459).


