Finding liminf of p_n/n where p_n = nth prime

[Dragonfall]

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).

 

[Gib Z]

You have to think what $$P_n$$~ $$n \log n$$ 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 $$\lim_{n\rightarrow {\infty}} \frac{P_n}{n\log n} = 1$$.

Since we want $$\lim_{n\rightarrow {\infty}} \frac{P_n}{n}$$, we can make the simple substitution and get log n.

 

[Dragonfall]

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.

 

[Gib Z]

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.

 

[Dragonfall]

An equivalent question is whether $$np_{n+1}-(n+1)p_n<0$$ for only finitely many n.