Finding liminf of p_n/n where p_n = nth prime

  • Thread starter Dragonfall
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  • #1
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Homework Statement



Find the lim inf of p_n/n where p_n is the nth prime.


Homework Equations



Well p_n ~ n logn, but I'm not sure if a simple substitution would work. This question may be incredibly trivial or open, and I can't figure out which.

I'm also wondering if the sequence above is monotone decreasing for sufficiently large n (this is not true for small n).
 
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Answers and Replies

  • #2
Gib Z
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You have to think what [tex]P_n [/tex]~ [tex]n \log n[/tex] actually means!

It means the quotient of the 2 functions as n approaches infinity is 1, note this does not mean the difference of the functions as n approaches infinity is 0.

eg n+2~n, but the difference of the functions will always be 2.

Anywho, so we know [tex]\lim_{n\rightarrow {\infty}} \frac{P_n}{n\log n} = 1[/tex].

Since we want [tex]\lim_{n\rightarrow {\infty}} \frac{P_n}{n}[/tex], we can make the simple substitution and get log n.
 
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  • #3
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Ok I wrote that backwards. I want n/p_n, not p_n/n. The question is whether lim n/(nlog n)=0 implies that lim inf n/p_n=0. More importantly is whether n/p_n is monotone for large n.
 
  • #4
Gib Z
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Well as per my previous post, the substitution is justified. So Your first question is true, it implies it. The 2nd part, I would not know.
 
  • #5
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An equivalent question is whether [tex]np_{n+1}-(n+1)p_n<0[/tex] for only finitely many n.
 

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