I am working through Apostol's Introduction To Analytic Number Theory, and am struggling to understand the proof to Lemma 7.5
The proof is laid out as follows: We begin with the sum $$ \sum_{n \leq x}\frac{\chi(n)\Lambda(n)}{n} $$ where $\Lambda(n)$ is Mangoldt's function, and express this sum in two ways. First we note that the definition of $\Lambda(n)$ gives us $$ \sum_{n \leq x}\frac{\chi(n)\Lambda(n)}{n} = \mathop{\sum_{p \leq x}\sum_{a = 1}^{\infty}}_{p^a \leq x}\frac{\chi(p^a)\ln(p)}{p^a} $$
We separate the terms with a = 1 and write $$ \sum_{n \leq x}\frac{\chi(n)\Lambda(n)}{n} = \sum_{p \leq x}\frac{\chi(p)\ln(p)}{p} + \mathop{\sum_{p \leq x}\sum_{a = 2}^\infty}_{p^a \leq x}\frac{\chi(p^a)\ln(p)}{p^a} $$
The second sum on the right is majorized by $$ \sum_p\ln(p)\sum_{a = 2}^\infty\frac{1}{p^a} = .... $$
My question is, what does it mean for a sum to be majorized, and how can I tell whether a sum is majorized or not, and if so, then by what.
"Majorized" means "bounded above" in this context. In particular, Apostol is bounding by absolute values, and bounding each Dirichlet character trivially $\chi(p^a) \leq 1$.