For Rosser's technique for proving incompleteness theorems, see Rosser's trick.

In number theory, Rosser's theorem states that the nth 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 nth 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).

See also

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\geq 2". Mathematics of Computation. 68 (225): 411–415. doi:10.1090/S0025-5718-99-01037-6. MR 1620223.