Currently I am reading Montgomery's paper on the Pair Correlation Conjecture. During the course of the proof of a lemma, he uses the the following estimation: if $x \geq 1$ and $1 < \sigma < 2$ and $t$ is any real then $$ \sum_{n = 1}^{\infty} \frac{x^{-2n}}{(\sigma - 1 - it - 2n)(\sigma + it + 2n)} = O_{\sigma} \left( \frac{x^{-3/2}}{|t|+2} \right). $$ Why does he uses this bound when he could have used the following slightly sharper bound $$ \sum_{n = 1}^{\infty} \frac{x^{-2n}}{(\sigma - 1 - it - 2n)(\sigma + it + 2n)} = O_{\sigma} \left( \frac{x^{-2}}{(|t|+2)^2} \right) $$
My bound proof: Note that $$ \frac{1}{(\sigma -1 - it -2n)(\sigma + it + 2n)}= O \left( \frac{1}{(|t|+2)^2} \right). $$ and so $$ \sum_{n = 1}^{\infty} \frac{x^{-2n}}{(\sigma -1 - it -2n)(\sigma + it + 2n)} = O \left( \frac{1}{(|t|+1)^2} \sum_{n=1}^{\infty} x^{-2n} \right) = O \left( \frac{x^{-2}}{(|t|+2)^2(1-x^{-2})} \right) = O \left( \frac{x^{-2}}{(|t|+1)^2} \right). $$
Is there any any subtlety I am overlooking here? Thank you for any guidance.