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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.

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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.).

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    $\begingroup$ a silly question : $\lambda_f(n)$ are the normalized coefs such that the critical strip of $\sum_n \lambda_f(n) n^{-s}$ is $Re(s) \in (0,1)$ (instead of $Re(s) \in (0,k)$) right ? $\endgroup$
    – reuns
    Commented Aug 30, 2016 at 18:38
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    $\begingroup$ @user1952009: Yes. $\endgroup$
    – GH from MO
    Commented Aug 30, 2016 at 18:50
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    $\begingroup$ and what is the idea for proving $L(s,f) = \sum_n \lambda_f(n) n^{-s}$ has no zero on $Re(s) = 1$, can we split $\log L(s,f)$ into $\sum_{m=1}^M C_mF_m(s) $ as we do for Dirichlet L-functions (where $F_m(s)$ are Dirichlet series with positive coefs and $e^{F_m(s)}$ are meromorphic and have no pole other than $s =1$) ? $\endgroup$
    – reuns
    Commented Aug 30, 2016 at 18:56
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    $\begingroup$ @user1952009: There are several methods. Perhaps the cleanest approach is to add to $\log L(s,f)$ a bunch of related series to form a Dirichlet series with nonnegative coefficients, and then use Landau's lemma. See, e.g., Section 3 in Moreno: Analytic proof of the strong multiplicity one theorem (American Journal of Mathematics 107 (1985), 163-206.). $\endgroup$
    – GH from MO
    Commented Aug 30, 2016 at 19:10

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