I am trying to read the following paper by Granville-Harper-Soundararajan and I had a few questions regarding the paper. They prove, for $S(x):=\sum_{n\leq x}f(n)$ and $F_x(s):=\prod_{p\leq x}\left(1+\prod_{p\leq x}\frac{f(p^k)}{p^{ks}}\right)$, the following theorem:
Suppose that $f$ is a multiplicative function with $|f(n)|\leq 1$, $$|S(x)|\ll x\frac{L(x)}{\log x}\log\left(100 \frac{\log x}{L(x)}\right)+x\frac{\log\log x}{\log x}\tag{1}$$
One of my questions is: why is this error term in $(1)$ better than the trivial bound $x$?
The second error term on the right, $x\frac{\log\log x}{\log x}$ is clearly better than $x$. But what about the left one? It seems, as the authors state, that $L(x)\leq 6 \log x$ for large $x$. However, that means that $x\frac{L(x)}{\log x}\log\left(100 \frac{\log x}{L(x)}\right)\ll x\log\left(100 \frac{\log x}{L(x)}\right)$. Call $L(x)/\log x$, $a(x)$. Then our error term is clearly $\ll x(\log 100-\log a(x))$, which to me does not seem to be better than $x$.
My second question is: One of the inequalities in the paper is: $$\sum_{x/pq\leq n \leq 2x/pq}\min\left(1, \frac{x}{T|x-pqn|}\right)\ll \left(1+\frac{x}{Tpq}\log T\right)\tag{2}$$ Here $p$ and $q$ are primes such that $p\in \mathcal{P_k}$ (as defined in the paper) and $q\leq 2x/p$. Further, $n$ is a natural number whose largest prime factor $\leq x$.
I was wondering how one could prove inequality $(2)$.
Maybe, it is something standard. Thank you for your time.
