Let $f$ be a primitive form of an even weight $k\geq 2$ for the full modular group $SL\_2(Z)$. Following from , Proposition 2.3 from 
[Z. Rudnick and P. Sarnak, Zeros of principal $L$-functions and random matrix theory, Duke Math. J. 81(1996), 269-322] and a standard
Riemann-Stieltjes partial integration, we plainly have
$$\sum\_{p\leq x} \frac{|\lambda\_f(p)|^2}{p}= \log{\log{x}}+O(1).$$
Is this identity is true:
$$\sum\_{z<p<w} \frac{|\lambda\_f(p)|^2}{p}=\sum\_{z<p<w} \frac{1}{p} + o(1),$$
provided that $z,$ $w$ are such that the RHS tends to infinity and $z$ tends to infinity.


Thanks in advance.



Relation between these two sums over prime numbers
Yes, and we don't even need that the RHS tends to infinity. If we perform the partial integration carefully (namely if we examine how the $O(1)$ term behaves at infinity), we find that
$$ \sum\_{p\leq x} \frac{|\lambda\_f(p)|^2}{p}= \log{\log{x}}+c\_f+O\left(\frac{1}{\log x}\right), $$
where $c\_f$ is a constant. Similarly, by Mertens' theorem,
$$ \sum\_{p\leq x} \frac{1}{p}= \log{\log{x}}+c+O\left(\frac{1}{\log x}\right), $$
where $c$ is a constant. Hence we get, under $z<w$ both tending to infinity,
$$ \sum\_{z<p<w} \frac{|\lambda\_f(p)|^2}{p}=\log\log w-\log\log z+o(1)=\sum\_{z<p<w} \frac{1}{p} + o(1).$$


**Added.** The error terms $O\left(\frac{1}{\log x}\right)$ can be improved to $O\left(e^{-k\sqrt{\log x}}\right)$ with a suitable $k>0$. For the second sum this is classical and follows from the Prime Number Theorem. For the first sum this was observed by Liu and Ye (American Journal of Mathematics 127 (2005), 837-849.).


