Rosser's theorem

In number theory, Rosser's theorem states that the nth prime number is greater than ${\displaystyle n\ln n}$. It was published by J. Barkley Rosser in 1939.[1]

Its full statement is:

Let p_n be the nth prime number. Then for n ≥ 1

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

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

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

See also

• Prime number theorem

References

1. 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
2. Dusart, Pierre (1999). "The kth prime is greater than k(log k + log log k−1) for k ≥ 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.