Rosser's theorem

For Rosser's technique for proving incompleteness theorems, see Rosser's trick.

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:

$p_{n}>n(\log n \log \log n-1).$

See also

• Prime number theorem

External links

• Rosser's theorem article on Wolfram Mathworld.