Rosser's theorem

In number theory, Rosser's theorem was published by J. Barkley Rosser in 1939. Its statement follows.

This article is about a theorem in number theory. For Rosser's technique for proving incompleteness theorems, see Rosser's trick. For Gödel–Rosser incompleteness theorems, see Gödel's incompleteness theorems. For the Church-Rosser theorem of λ-calculus, see Church-Rosser theorem.

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

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

This result was subsequently improved upon to be[1]:

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

See also

• Prime number theorem

References

1. 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.

• Rosser, J. B. "The n-th Prime is Greater than n log n". Proceedings of the London Mathematical Society 45, 21-44, 1939.

External links

• Rosser's theorem article on Wolfram Mathworld.