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 for $n \geq 2$ :

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

See also

Prime number theorem

References

↑ Rosser, J. B. "The $n$ -th Prime is Greater than $n \log n$ ". Proceedings of the London Mathematical Society 45:21-44, 1939. doi:10.1112/plms/s2-45.1.21
↑ Dusart, Pierre (1999). "The $k$ th prime is greater than $k (\log k + \log \log k - 1)$ for $k \geq 2$ ". Mathematics of Computation. 68 (225): 411–415. doi:10.1090/S0025-5718-99-01037-6. MR 1620223.

External links

Rosser's theorem article on Wolfram Mathworld.