The $k^{th}$ prime is greater than $k(\ln k +\ln\ln k -1)$ for $k\geq 2$

ROSSER and SCHOENFELD have used the fact that the first 3,500,000 zeros of the RIEMANN zeta function lie on the critical line to give estimates on $\psi(x)$ and $\theta(x)$. With an improvement of the above result by BRENTet al., we are able to improve these estimates and to show that the $k^{th}$ prime is greater than $k(\ln k +\ln\ln k -1)$ for $k\geq 2$. We give further results without proof.