Hi all,
I'm reading the article of Murty & Murty "An analogue of Erdös-Kac theorem for fourier coefficients of modular forms"
There is a part of the article that I don't understand.
Briefly,
Let $g$ be an integer valued multiplicative function satisfying the following conditions
1) $|g(n)| \le n^{\beta}$ for some $\beta$ positive
2) if $ \pi (x;g,d)={p \le x, g(p) \equiv 0 (d)}$, then, $\sum_{d \leq x^{\theta}}{|\pi (x;g,d)- \delta(d)\pi(x)|} \ll \pi(x)$ for some $\theta$ positive
where
$$\delta(p^a)=\frac{1}{p^a}+O(\frac{1}{p^{a+1}})$$
and
$$\delta(p^aq^b)=(\frac{1}{p^a}+O(\frac{1}{p^{a+1}}))(\frac{1}{q^b}+O(\frac{1}{q^{b+1}}))$$
for distinct primes.
Here is the part I don't understand
They somehow obtained the following bound
$$\sum_{q \leq x^{\beta}, \alpha \geq 2} \pi(x;g, q^{\alpha}) \ll \frac{x}{\log x}$$
If I split the sum into two parts $q^{\alpha} \leq x^{\theta}$ and $q^{\alpha} > x^{\theta} $, then I'm able to control the first sum but the second sum remains problematic.
Any Suggestions?
Best
Rahmi
