As $T\to\infty$, we have
$N(T) = \frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+\frac{7}{8}+\frac{1}{\pi}\arg\zeta(\frac{1}{2}+iT)+O(T^{-1})$.
This is proved in Chapter 15 of Davenport's Multiplicative Number Theory. The error term $O(T^{-1})$ is essentially truncation for the asymptotic expansions for the arctan and gamma functions. The contribution from the arctan function consists of lower order terms in a Taylor expansion, and the contribution from the gamma function consists of lower order terms in the Stirling expansion.