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 \[ | N ( T ) − T 2 π log ⁡ ( T 2 π e ) | ≤ 0.10076 log ⁡ T + 0.24460 log ⁡ log ⁡ T + 8.08344 , | N (T) - T/ 2 π log ( T/2π e ) |≤ 0.10076log T+0.24460log log T+8.08344, \] for every T ≥ e T≥ e , where N ( T ) N(T) is the number of non-trivial zeros ρ = β + i γ ρ =β +iγ , with 0 > γ ≤ T 0>γ ≤ T , of the Riemann zeta-function ζ ( s ) ζ (s) . The main source of improvement comes from implementing new subconvexity bounds for ζ ( σ + i t ) ζ (σ +it) on some