A Shorter Proof of Rosser's Theorem Without Using The Prime Number Theorem

Question

While researching on the elementary proof of Bertrand's Postulate I came to know about a theorem of Rosser's which states that $p_n$ $>$ $n$ $\text{ln}$ $n$. I have seen Rosser's original proof and I am interested in shorter proof of the result. It will be best if it could be proved using elementary methods. Any suggestion will be appreciated.

Answer 1

There is an elementary proof by Chebyshev, which shows that $p_n>\alpha n\log(n)$ for a constant $a<1$ close to $1$. However, for $\alpha=1$ one needs more, i.e., one needs the techniques of Rosser and Schoenfeld, using zero-free regions for the Riemann zeta function. I think there is no easier proof than the one of Rosser. Piere Dusart has sharpened the estimates on the $n$-th prime. This is perhaps easier to read. References can be found in the answers here.