This query is related to the one here, but it is on justifying the mathematics done to bound the term.
Extracting the stage of analysis that I am trying to get my head around, Apostol claims that the double sum $$\mathop {\sum\limits_{p\,\, \le \,\,x} {\sum\limits_{a = 2}^\infty {} } }\limits_{{p^a} \le x} \,\frac{{\chi ({p^a})\log p}}{{{p^a}}}$$
can be bounded above by
$$\sum\limits_p {\log p\,} \sum\limits_{a = 2}^\infty {\frac{1}{{{p^a}}}} $$
The one thing that disturbs me about this is the fact that the Dirichlet characters ${\chi ({p^a})}$, are, in general, complex roots of unity.
How can it be justified that summations involving complex quantities can be bounded with real numbers?
Surely the sum that generates the above analysis, $$\sum\limits_{n\,\, \le x} {\frac{{\chi (n)\Lambda (n)}}{n}} $$ will also, in general, be complex.
Apostol, in this "Introduction to Analytic Number Theory" (page 151) does not make any mention of this.
What am I missing?
Thanks for any help/insight.