The 𝑘^{𝑡ℎ} prime is greater than 𝑘(ln𝑘+lnln𝑘-1) for 𝑘≥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 ψ ( x ) \psi (x) and θ ( x ) \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 t h k^{th} prime is greater than k ( ln ⁡ k + ln ⁡ ln ⁡ k − 1 ) k(\ln k +\ln \ln k -1) for k ≥ 2 k\geq 2 . We give further results without proof.