  In this article, we show that $$ \left| N (T) - \frac{T}{ 2 \pi} \log \left(
\frac{T}{2\pi e}\right) \right|
  \le 0.1038 \log T + 0.2573 \log\log T + 9.3675 $$ where $N(T)$ denotes the
number of non-trivial zeros $\rho$, with $0<\Im(\rho) \le T$, of the Riemann
zeta function. This improves the previous result of Trudgian for sufficiently
large $T$. The improvement comes from the use of various subconvexity bounds
and ideas from the work of Bennett $et$ $al.$ on counting zeros of Dirichlet
$L$-functions.
