There exists Lemma giving that, denot $\rho(n)$ as some multiplicative function, and I want to consider only for the main term $$\sum_{\substack{M<m<M_1 \\ \text{some other conditions}}}\rho(m) \approx \dfrac{\pi}{4}(M_1-M)A(q)$$
It is said (from the excerpt that I tried to catch up) that by using the lemma with partial summation $$\sum_{\substack{M<m<2M \\ \text{some other conditions}}}\dfrac{\rho(m)}{m^2}\approx \dfrac{\pi}{8}M^{-1}A(n_1n_2)$$ but as I take a calculation $$\sum_{\substack{M<m<2M \\ \text{some other conditions}}}\dfrac{\rho(m)}{m^2}=\sum_{\substack{M<m<2M \\ \text{some other conditions}}}\rho(m)\cdot\dfrac{1}{m^2}\approx \left(\sum_{\substack{M<m<2M \\ \text{some other conditions}}}\rho(m)\right)\dfrac{1}{4M^2}\approx\dfrac{\pi}{16}M^{-1}A(n_1n_2)$$ so I don't know why the constant is $1/8$ instead of $1/16$.