Rosser's theorem

{{Short description|The nth prime number exceeds n log(n).}}

{{for|Rosser's technique for proving incompleteness theorems|Rosser's trick}}

In number theory, Rosser's theorem states that the $n$th prime number is greater than $n\; \backslash log\; n$, where $\backslash log$ is the natural logarithm function. It was published by J. Barkley Rosser in 1939.Rosser, J. B. "The $n$-th Prime is Greater than $n\backslash log\; n$". Proceedings of the London Mathematical Society 45:21-44, 1939. {{doi|10.1112/plms/s2-45.1.21}}{{Closed access}}

Its full statement is:

Let $p\_n$ be the $n$th prime number. Then for $n\backslash geq\; 1$

:$p\_n\; >\; n\; \backslash log\; n.$

In 1999, Pierre Dusart proved a tighter lower bound:{{cite journal|authorlink=Pierre Dusart|last=Dusart|first=Pierre|title=The $k$th prime is greater than $k(\backslash log\; k\; +\; \backslash log\backslash log\; k\; -\; 1)$ for $k\backslash geq\; 2$|journal=Mathematics of Computation|volume=68|issue=225|year=1999|pages=411–415|mr=1620223|doi=10.1090/S0025-5718-99-01037-6|doi-access=free}}

:$p\_n\; >\; n\; (\backslash log\; n\; +\; \backslash log\; \backslash log\; n\; -\; 1).$