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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.

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:

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

See also

• Prime number theorem

References

1. Rosser, J. B. "The ${\displaystyle n}$-th Prime is Greater than ${\displaystyle n\log n}$". Proceedings of the London Mathematical Society 45:21-44, 1939. doi:10.1112/plms/s2-45.1.21
2. Dusart, Pierre (1999). "The ${\displaystyle k}$th prime is greater than ${\displaystyle k(\log k+\log \log k-1)}$ for ${\displaystyle k\geq 2}$". 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.

• Theorems about prime numbers