Theorem 13.4 Apostol, pg 285 - unnecessary condition?

Question

In Theorem 13.4:

Why do we need $\sigma \ge \frac{1}{2}$? The representation for $\zeta(s)$ holds for all $\sigma > 0$ in (17) (page 284) and the proof does not utilize $\sigma \ge \frac{1}{2}.$

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EDIT: Actually, I believe it is used in the line, $$ \frac{2t}{\sigma N^{\sigma}} = O( \frac{1}{N} ). $$