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As $T\to\infty$, we have $$ N(T) = \frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+\frac{7}{8}+\frac{1}{\pi}\arg\zeta\left(\frac{1}{2}+iT\right)+O(T^{-1}). $$ This is proved in Chapter 15 of Davenport's Multiplicative Number Theory. The error term $O(T^{-1})$ is a 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.

As $T\to\infty$, we have $$ N(T) = \frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+\frac{7}{8}+\frac{1}{\pi}\int_{\frac{1}{2}}^{\infty}\mathrm{Im}\Big(-\frac{\zeta'}{\zeta}(\sigma+iT)\Big)d\sigma+O(T^{-1}). $$ This is proved in Chapter 15 of Davenport's Multiplicative Number Theory. The error term $O(T^{-1})$ is a 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.

EDIT: Because of the apparent lack of clarity regarding "arg" in this result, I replaced "$\arg \zeta(1/2+iT)$" with the corresponding integral, which should not be ambiguous. I hope this helps.

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 a 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.

As $T\to\infty$, we have $$ N(T) = \frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+\frac{7}{8}+\frac{1}{\pi}\arg\zeta\left(\frac{1}{2}+iT\right)+O(T^{-1}). $$ This is proved in Chapter 15 of Davenport's Multiplicative Number Theory. The error term $O(T^{-1})$ is a 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.

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.

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 a 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.