I am studying the paper M. Ram Murty, V. Kumar Murty: Mean values of derivatives of modular $L$-series, Ann. of Math. (2) 133 (1991), no. 3, 447-475.


Let $L(s)=\sum\_{n=1}^{\infty} \frac{a(m)}{m^s}$ be the $L$-function of a modular elliptic curve with conductor $N$. We decompose $m=m\_1m\_2$ such that $m\_1$ is the product of prime factors (with multiplicity) of $m$ which divide $N$ (i.e., $m\_2=n/m\_1$ is coprime to $N$). We only consider $m$ such that $m\_2$ is square, so we write $m=m\_1m\_2^2$ instead of $m=m\_1m\_2$.


In p.457, the authors state that


$$Y\sum\_{m < \frac{1}{2}X}\frac{|a(m\_1m\_2^2)|}{m\_1m\_2^2} \log\frac{X}{m}+Y\sum\_{m>\frac{1}{2}X}\frac{|a(m\_1m\_2^2)|}{m\_1m\_2^2}\tau(m)\mathrm{exp}(-m/X) \ll Y(\log X)^2.$$


However, I don't know why this estimate holds.



Upper bound of summation $\sum_{m < \frac{1}{2}X} \frac{|a(m_1m_2^2)|}{m_1m_2^2} \log\frac{X}{m}$
Here is a proof for non CM elliptic curves. In this case, the Dirichlet series
$$\sum\_{n=1}^\infty\frac{a(n^2)^2}{n^{2+s}}$$
is holomorphic in the closed half-plane $\Re(s)\geq 1$ with a simple pole at $s=1$. This follows, for example, from Lemma 3 in Moreno-Shahidi: The fourth moment of Ramanujan $\tau$-function (1983) combined with Lemma 2.2 in Lü: Sums of absolute values of cusp form coefficients and their application (2014). Of course Lü's paper is more recent than that of Murty and Murty, but the quoted lemma is rather standard. By the Wiener-Ikehara theorem, we conclude the asymptotic formula (for some $c>0$)
$$\sum\_{n\leq T}\frac{a(n^2)^2}{n^2}\sim c T\qquad\text{as}\qquad T\to\infty.$$
In particular,
$$\sum\_{T/2<n\leq T}|a(n^2)|\leq T\sum\_{T/2<n\leq T}\frac{|a(n^2)|}{n}\leq T\sum\_{T/2<n\leq T}\left(1+\frac{a(n^2)^2}{n^2}\right)\ll\_N T^2.$$
From here the bound follows easily by a dyadic decomposition in $m\_2$, upon noting that
$$\sum\_{m\_1\mid(2N)^\infty}\frac{|a(m\_1)|\tau(m\_1)}{m\_1}\ll\_N 1.$$


**P.S.** See my comments below for additional details.


