EDITED TO REFLECT @LUCIA'S COMMENTS: After digging through a bunch of references, I sorted out the answer I was looking for. For all $T > 0$ except at the points of discontinuity of $N(2\pi T)$, one has $$N(2 \pi T) =1+ \frac{1}{\pi} \theta(2\pi T) + \frac{1}{\pi}\operatorname{Arg}\zeta\left(\frac{1}{2}+2\pi i T\right)+R(T),$$ where $\operatorname{Arg}$ is the principal branch of the argument, where $\theta(T)$ is the Riemann-Siegel theta function, and where $R(T)$ is an integer that is $O(\log T)$. One has the asymptotic relation \begin{align*} N(2 \pi T) & =T \log T-T+\frac{7}{8}+\frac{1}{\pi}\operatorname{Arg}\zeta\left(\frac{1}{2}+2\pi i T\right)+R(T)+O\left(\frac{1}{T}\right) \\ & = \int_1^T \log t \, dt -\frac{1}{8}+\frac{1}{\pi}\operatorname{Arg}\zeta\left(\frac{1}{2}+2\pi i T\right)+R(T)+O\left(\frac{1}{T}\right) \end{align*} and the asymptotic expansion \begin{align*} N(2 \pi T) - \frac{1}{\pi}\operatorname{Arg}\zeta\left(\frac{1}{2}+2\pi i T\right) \sim T \log T - T+ R(T) + \frac{7}{8}+\frac{1}{96 \pi^2 T}+ \frac{7}{11340\pi^4 T^3}+ \frac{31}{161280 \pi^6 T^5}+\cdots \end{align*} as $T \to \infty$, where the numerators and (1/2)denominators are as in OEIS Sequences A036282 and A114721, respectively. Here is a plot of the function $N(2\pi T)$ and its smooth approximation $1+\frac{1}{\pi}\theta(2\pi T)$.
What is the infimum of all $T$ that is not $\frac{1}{2\pi}$ of an imaginary part of a zero of $\zeta(s)$ such that $R(T)\neq 0$?
