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.^[1]

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:^[2]

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

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.

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