Rosser's theorem

In number theory, Rosser's theorem states that the $$n$$th prime number is greater than $$n \log n $$, where $$\log$$ is the natural logarithm function. It was published by J. Barkley Rosser in 1939.

Its full statement is:

Let $$p_n$$ be the $$n$$th prime number. Then for $$n\geq 1$$


 * $$p_n > n \log n. $$

In 1999, Pierre Dusart proved a tighter lower bound:


 * $$ p_n > n (\log n + \log \log n - 1). $$