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.