In addition to 2734364041's answer, the following paper of Tim Trudgian may be useful:

https://www.sciencedirect.com/science/article/pii/S0022314X13002047

In particular, Trudgian shows that for all $T\geq e$,

$$\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)-\frac{7}{8}\right|\leq 0.112\log T+0.278\log\log T+2.510+\frac{0.2}{T}$$

and also includes a useful discussion of all the terms that go into this result.

In addition to 2734364041's answer, this paper of Tim Trudgian may be useful: in particular, Trudgian shows that for all $T\geq e$,

$$\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)-\frac{7}{8}\right|\leq 0.112\log T+0.278\log\log T+2.510+\frac{0.2}{T}$$

and also includes a useful discussion of all the terms that go into this result.

In addition to 2734364041's answer, the following paper of Tim Trudgian may be useful:

https://www.sciencedirect.com/science/article/pii/S0022314X13002047

In particular, Trudgian shows that for all $T\geq e$,

$$\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi}\right)-\frac{7}{8}\right|\leq 0.112\log T+0.278\log\log T+2.510+\frac{0.2}{T}$$

and also includes a useful discussion of all the terms that go into this result.

In addition to 2734364041's answer, the following paper of Tim Trudgian may be useful:

https://www.sciencedirect.com/science/article/pii/S0022314X13002047

In particular, Trudgian shows that for all $T\geq e$,

$$\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)-\frac{7}{8}\right|\leq 0.112\log T+0.278\log\log T+2.510+\frac{0.2}{T}$$

and also includes a useful discussion of all the terms that go into this result.