Improved estimates for the argument and zero-counting function of the Riemann zeta-function

In this article, we improve the recent work of Hasanalizade, Shen, and Wong by establishing $$\mid N(T)-\frac{T}{2\pi }\log \left(\frac{T}{2\pi e}\right)\mid \le 0.10076\log T+0.24460\log \log T+8.08344,$$ for every $T\ge e$, where $N(T)$ is the number of non-trivial zeros $\rho =\beta +i\gamma$, with $0<\gamma \le T$, of the Riemann zeta-function $\zeta (s)$. The main source of improvement comes from implementing new subconvexity bounds for $\zeta (\sigma +it)$ on some ${\sigma }_k$-lines inside the critical strip.