Rosser's theorem
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In number theory, Rosser's theorem states that the [math]\displaystyle{ n }[/math]th prime number is greater than [math]\displaystyle{ n \log n }[/math], where [math]\displaystyle{ \log }[/math] is the natural logarithm function. It was published by J. Barkley Rosser in 1939.[1]
Its full statement is:
Let [math]\displaystyle{ p_n }[/math] be the [math]\displaystyle{ n }[/math]th prime number. Then for [math]\displaystyle{ n\geq 1 }[/math]
- [math]\displaystyle{ p_n \gt n \log n. }[/math]
In 1999, Pierre Dusart proved a tighter lower bound:[2]
- [math]\displaystyle{ p_n \gt n (\log n + \log \log n - 1). }[/math]
See also
References
- ↑ Rosser, J. B. "The [math]\displaystyle{ n }[/math]-th Prime is Greater than [math]\displaystyle{ n\log n }[/math]". Proceedings of the London Mathematical Society 45:21-44, 1939. doi:10.1112/plms/s2-45.1.21
- ↑ Dusart, Pierre (1999). "The [math]\displaystyle{ k }[/math]th prime is greater than [math]\displaystyle{ k(\log k + \log\log k - 1) }[/math] for [math]\displaystyle{ k\geq 2 }[/math]". Mathematics of Computation 68 (225): 411–415. doi:10.1090/S0025-5718-99-01037-6.
External links
- Rosser's theorem article on Wolfram Mathworld.
de:John Barkley Rosser#Satz von Rosser
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