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

Abstract:

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 Brent et 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.