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- P. Dusart, "The kth prime is greater than k(ln k+ln ln k-1) for k> 2," Math. Comp., 68:225 (January 1999) 411--415. MR 99d:11133
Rosser and Schoenfeld have use 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) and θ(x). With an improvement of the above result by Brent et. al., we are able to improve these estimates and to show that the kth prime is greater than k(ln k + ln ln k -1) for k ≥ 2. We give further results without proof.