Eq. 2.1 in Levinson's paper states $$\dfrac{1}{(s-1)^m} - \sum_{n=0}^{\infty} \dfrac{1}{(s+2n)^m} - \sum_{\rho} \dfrac{1}{(s-\rho)^m} = - \sum_{|\gamma -t|<1} \dfrac{1}{(s-\rho)^m} + \mathcal{O} (\log t).$$ And it is stated that this is derived by $N(T+1) - N(T) = \mathcal{O} (\log t)$, where $N(T)$ are the number of zeros of $\zeta(\sigma + it)$ for $0 < t \le T$.
My question is how to derive Eq. 2.1 rigorously? Roughly speaking $\dfrac{1}{(s-1)^m}$ and $\sum_{n=0}^{\infty} \dfrac{1}{(s+2n)^m}$ and $\sum_{|\gamma -t| \ge 1} \dfrac{1}{(s-\rho)^m}$ are $\mathcal{O} (1)$ (?) and thus are dominated by $\mathcal{O} (\log t)$. If so, then what is the use of $N(T+1) - N(T) = \mathcal{O} (\log t)$ (as the paper claims)?