Rosser's theorem

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

In number theory, Rosser's theorem states that the ${\displaystyle n}$th prime number is greater than ${\displaystyle n\log n}$, where ${\displaystyle \log }$ is the natural logarithm function. It was published by J. Barkley Rosser in 1939.^[1]

Its full statement is:

Let ${\displaystyle p_{n}}$ be the ${\displaystyle n}$th prime number. Then for ${\displaystyle n\geq 1}$

${\displaystyle p_{n}>n\log n.}$

In 1999, Pierre Dusart proved a tighter lower bound:^[2]

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

See also

• Prime number theorem